Matrices Questions

We provide matrices practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on matrices skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

List of matrices Questions

Question No Questions Class
1 ( fleft(begin{array}{ccc}-1 & 2 & 4 \ 3 & 6 & -5end{array}right] ) then find ( 3 A ) 12
2 The element in the first row and third
column of the inverse of the matrix
( left[begin{array}{ccc}1 & 2 & -3 \ 0 & 1 & 2 \ 0 & 0 & 1end{array}right] ) is
A . -2
B.
( c cdot 1 )
D. 7
12
3 Trace of matrix ( boldsymbol{A}^{k} ) is
( mathbf{A} cdot 3^{k}+1+(-1)^{k} )
B . ( 2^{k}+3^{k}-2 )
c. ( 3^{k}-2^{k}+2 )
( mathbf{D} cdot 2^{k}+1 )
12
4 If ( A=left[begin{array}{ccc}1 & 2 & 3 \ 2 & 3 & 4 \ 0 & 5 & 6end{array}right], ) then ( 2 A= )
A. ( left[begin{array}{lll}2 & 4 & 6 \ 2 & 3 & 4 \ 0 & 5 & 6end{array}right] )
в. ( left[begin{array}{lll}1 & 2 & 3 \ 4 & 6 & 8 \ 0 & 5 & 6end{array}right] )
с. ( left[begin{array}{lll}1 & 2 & 3 \ 2 & 3 & 4 \ 0 & 10 & 12end{array}right] )
D. ( left[begin{array}{lll}2 & 4 & 6 \ 4 & 6 & 8 \ 0 & 10 & 12end{array}right] )
12
5 If ( boldsymbol{A}=left[begin{array}{ll}2 & 3 \ 5 & 7end{array}right], B=left[begin{array}{cc}0 & 4 \ -1 & 7end{array}right], C= )
( left[begin{array}{cc}1 & 0 \ -1 & 4end{array}right], ) find ( A C+B^{2}-10 C )
12
6 Q Type your question
( left.begin{array}{lll}mathbf{0} & mathbf{0} & mathbf{1}end{array}right], mathbf{i} mathbf{z}- )
( left[begin{array}{lll}mathbf{1} & mathbf{0} & mathbf{0} \ mathbf{0} & mathbf{0} & mathbf{1} \ mathbf{0} & mathbf{1} & mathbf{0}end{array}right], boldsymbol{P}_{mathbf{3}}=left[begin{array}{lll}mathbf{0} & mathbf{1} & mathbf{0} \ mathbf{1} & mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{0} & mathbf{1}end{array}right] )
( boldsymbol{P}_{4}=left[begin{array}{ccc}0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 0 & 1end{array}right], P_{5}= )
( left[begin{array}{lll}0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0end{array}right], P_{6}=left[begin{array}{lll}0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0end{array}right], ) and
( boldsymbol{X}=sum_{boldsymbol{k}=1}^{6} boldsymbol{P}_{boldsymbol{K}}left[begin{array}{lll}2 & 1 & 3 \ 1 & 0 & 2 \ 3 & 2 & 1end{array}right] boldsymbol{P}_{K}^{T} )
Where ( P_{K}^{T} ) denotes the transpose of
matrix ( P_{k} ). Then which of the following
option is / are correct?
This question has multiple correct options
A. ( X ) is symmetric matrix
B. The sum of the diagonal entries of ( X ) is 18
c.
If ( Xleft[begin{array}{l}1 \ 1 \ 1end{array}right]=alphaleft[begin{array}{l}1 \ 1 \ 1end{array}right] ) then ( alpha=30 )
D. ( X-30 I ) is an invertible matrix
12
7 Assertion ( mathbf{f}[boldsymbol{x} mathbf{1}]left[begin{array}{cc}mathbf{1} & mathbf{0} \ -mathbf{2} & mathbf{3}end{array}right]left[begin{array}{c}boldsymbol{x} \ -mathbf{5}end{array}right]=mathbf{0}, ) then value
of ( x ) is either- 3 or 5
Reason Two matrices ( left[begin{array}{ll}boldsymbol{x} & boldsymbol{y} \ boldsymbol{u} & boldsymbol{v}end{array}right] ) & ( left[begin{array}{ll}boldsymbol{a} & boldsymbol{b} \ boldsymbol{c} & boldsymbol{d}end{array}right] ) are
equal if ( & ) only if their corresponding
entries are equal
& only if their corresponding entries are equal
A. Both (A) & (R) are individually true & (R) is correct explanation of (A)
B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
c. (A)is true but (R) is false
D. (A)is false but (R) is true
12
8 Suppose ( A ) and ( B ) are two square
matrices of same order. If ( boldsymbol{A}, boldsymbol{B} ) are
symmetric matrices, then ( A B-B A ) is
A. A symmetric matrix
B. A skew symmetric
c. A scalar matrix
D. A triangular matrix
12
9 (a) a = 2ab, B = a? +62
(©) a = 2? +B2, B= 2ab
(b) a = a? +62, B = ab
(d) a=a? +62, B=a? –62.
12
10 Find the inverse of ( boldsymbol{A}=left[begin{array}{lll}mathbf{0} & mathbf{1} & mathbf{2} \ mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{3} & mathbf{1} & mathbf{1}end{array}right] )
( boldsymbol{A}^{-1}=left[begin{array}{ccc}1 / 2 & -1 / 2 & 1 / 2 \ a & 3 & b \ c & -3 / 2 & 1 / 2end{array}right] )
Find ( |boldsymbol{a} boldsymbol{b} boldsymbol{c}| ? )
12
11 Find the order of the following matrices.
(i) ( left.begin{array}{ccc}1 & -1 & 5 \ -2 & 3 & 4end{array}right) )
(ii)
(iii) ( begin{array}{rl}3 & -26 \ 6 & -11 \ 2 & 4end{array} )
(iv) (345)
( (v)left[begin{array}{cc}1 & 2 \ -2 & 3 \ 9 & 7 \ 6 & 4end{array}right] )
12
12 If ( boldsymbol{A}=left[begin{array}{cc}1 & 2 \ 3 & 4 \ 5 & 6end{array}right] ) and ( B=left[begin{array}{cc}-3 & -2 \ 1 & -5 \ 4 & 3end{array}right] )
then find ( D=left[begin{array}{ll}boldsymbol{p} & boldsymbol{q} \ boldsymbol{r} & boldsymbol{s} \ boldsymbol{t} & boldsymbol{u}end{array}right] ) such that ( boldsymbol{A}+ )
( boldsymbol{B}-boldsymbol{D}=boldsymbol{O} )
( mathbf{A} cdotleft[begin{array}{cc}-1 & 0 \ 4 & -1 \ 9 & 9end{array}right] )
( mathbf{B} cdotleft[begin{array}{cc}-3 & 0 \ 4 & -1 \ 9 & 9end{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}-2 & 0 \ 4 & -1 \ 9 & 9end{array}right] )
D. ( left[begin{array}{cc}-2 & 0 \ 4 & -5 \ 9 & 9end{array}right] )
12
13 Let ( quad A=left(begin{array}{ccc}x^{2} & 6 & 8 \ 3 & y^{2} & 9 \ 4 & 5 & z^{2}end{array}right) ) and ( B= )
( left(begin{array}{ccc}2 x & 3 & 5 \ 2 & 2 y & 6 \ 1 & 4 & 2 z-3end{array}right) ) be two matrices
and if ( operatorname{Tr}(boldsymbol{A})=boldsymbol{T} boldsymbol{r}(boldsymbol{B}), ) then the value of
( (x+y+z) ) is equal to
(Note: ( operatorname{Tr}(P) ) denotes trace of matrix ( P ) )
( A )
B.
( c )
( D )
12
14 Which one of the following is true for any two square matrices ( A ) and ( B ) of
same order?
( mathbf{A} cdot(A B)^{T}=A^{T} B^{T} )
B . ( left(A^{T} Bright)^{T}=A^{T} B^{T} )
c. ( (A B)^{T}=B A )
D. ( (A B)^{T}=B^{T} A^{T} )
12
15 A matrix has 8 elements. What are the
possible orders it can have?
12
16 If ( boldsymbol{A} ) satisfies the equation ( boldsymbol{x}^{3}-mathbf{5} boldsymbol{x}^{2}+ )
( 4 x+k I=0, ) then ( A^{-1} ) exists if
A. ( k neq-1 )
в. ( k neq 0 )
c. ( k neq 1 )
D. none of these
12
17 If ( A ) is a skew-symmetric matrix, then
trace of ( boldsymbol{A} ) is
12
18 If ( A=left[begin{array}{cr}2 & 5 a \ -3 & 1end{array}right] ) and ( A ) doesn’t have multiplicative inverse then find ( mathbf{A} ) 12
19 For what value of ( x, ) is the matrix ( A= ) ( left[begin{array}{ccc}mathbf{0} & mathbf{1} & mathbf{- 1} \ -mathbf{1} & mathbf{0} & mathbf{3} \ boldsymbol{x} & mathbf{- 3} & mathbf{0}end{array}right] ) a skew symmetric
matrix?
12
20 matrix is a square matrix in
which all the elements other than the
principal diagonal elements are zero.
A. scalar
B. null
c. diagonal
D. unit
12
21 ( left|begin{array}{cc}mathbf{2} & mathbf{3} \ mathbf{4} & mathbf{4}end{array}right|+left|begin{array}{cc}boldsymbol{x} & mathbf{3} \ boldsymbol{y} & mathbf{1}end{array}right|=left|begin{array}{cc}mathbf{1 0} & mathbf{6} \ mathbf{8} & mathbf{5}end{array}right|, ) then ( (mathbf{x}, mathbf{y})= )
A ( cdot(4,8) )
( B cdot(8,4) )
( c cdot(1,2) )
( D cdot(2,4) )
12
22 If ( mathbf{A}=left[mathbf{a}_{mathbf{i} mathbf{j}}right] ) is a scalar matrix of order
( boldsymbol{n} times boldsymbol{n} ) such that ( mathbf{a}_{mathbf{i j}}=mathbf{k} ) for all ( mathbf{i}=boldsymbol{j}, ) then
trace of ( mathbf{A}= )
( A cdot ) nk
B. ( n+k )
c. ( frac{n}{k} )
D.
12
23 Let ( boldsymbol{A}=left[begin{array}{cc}mathbf{2} & mathbf{3} \ mathbf{- 1} & mathbf{5}end{array}right] cdot ) If ( boldsymbol{A}^{-mathbf{1}}=boldsymbol{x} boldsymbol{A}+boldsymbol{y} boldsymbol{I} )
find ( boldsymbol{x}+mathbf{2} boldsymbol{y} )
12
24 cos
-sino
41.
IfA= sino
cose ]
|, then the matrix A-50
when 0 = *, is equal to:
[JEE M 2019-9 Jan (M)]
(b)
1
V3
ISO-IN
(d)
v3
12
25 Find the value of ( y-x ) from the
following equation ( mathbf{2}left[begin{array}{cc}boldsymbol{x} & mathbf{5} \ mathbf{7} & boldsymbol{y}-mathbf{3}end{array}right]+left[begin{array}{cc}mathbf{3} & mathbf{- 4} \ mathbf{1} & mathbf{2}end{array}right]=left[begin{array}{cc}mathbf{7} & mathbf{6} \ mathbf{1 5} & mathbf{1 4}end{array}right] )
12
26 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{3} & -mathbf{4} \ mathbf{1} & -mathbf{1}end{array}right], ) then prove that ( boldsymbol{A}- )
( A^{T} ) is a skew-symmetric matrix.
12
27 For what value of ( x, ) is the matrix ( A= )
( left[begin{array}{ccc}mathbf{0} & mathbf{1} & -mathbf{2} \ -mathbf{1} & mathbf{0} & mathbf{3} \ boldsymbol{x} & mathbf{- 3} & mathbf{0}end{array}right] ) a skew-symmetric
matrix?
12
28 ( mathbf{f}left[begin{array}{ccc}mathbf{1} & mathbf{3} & mathbf{0} \ mathbf{1} & mathbf{0} & -mathbf{2} \ mathbf{- 4} & mathbf{- 4} & mathbf{4}end{array}right]=mathbf{A}+mathbf{B} ) where ( mathbf{A} )
is symmetric matrix and B is skew-
symmetric, then ( mathbf{A}-mathbf{B} ) is equal to
( mathbf{A} cdotleft[begin{array}{ccc}1 & 1 & -4 \ 3 & 0 & -4 \ 0 & -2 & 4end{array}right] )
B. ( left[begin{array}{lll}2 & 1 & 3 \ -1 & 2 & 4 \ 3 & -1 & 2end{array}right] )
( mathbf{c} cdotleft[begin{array}{lll}0 & 1 & -1 \ 2 & 3 & 4 \ -4 & 1 & 2end{array}right] )
D. ( left[begin{array}{ccc}2 & -3 & 0 \ 0 & 1 & 2 \ 2 & 4 & 0end{array}right] )
12
29 In the matrix, write:
( A=left[begin{array}{ccc}2 & 5 & 19-7 \ 35-2 & frac{5}{2} & 12 \ sqrt{3} & 1 & -517end{array}right] )
(i) The order of the matrix
(ii) The number of elements
(iii) Write the elements ( a_{13}, a_{21}, a_{33}, a_{24}, a_{23} )
12
30 Assertion
The possible dimension of a matrix
consisting 27 elements is 4
Reason
The number of ways of expressing 27 as a product of two positive integers is 4
A. Both Assertion & Reason are individually correct & Reason is correct explanation of Assertion,
B. Both Assertion & Reason are individually true but Reason is Not the correct explanation of Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct.
12
31 If ( boldsymbol{A}=left|begin{array}{c}-mathbf{2} \ mathbf{4} \ mathbf{5}end{array}right|, boldsymbol{B}=|mathbf{1} quad mathbf{3}-mathbf{6}|, ) State
whether ist is true or false ( (A B)^{1}= )
( boldsymbol{B}^{1} boldsymbol{A}^{1} )
A. True
B. False
12
32 If ( mathbf{A}=left[begin{array}{cc}mathbf{2} & mathbf{2} \ -mathbf{3} & mathbf{2}end{array}right], mathbf{B}=left[begin{array}{cc}mathbf{0} & mathbf{- 1} \ mathbf{1} & mathbf{0}end{array}right] ) then
( mathbf{B}^{-mathbf{1}} mathbf{A}^{-mathbf{1}} )
A ( cdot frac{1}{10}left[begin{array}{cc}2 & 2 \ -3 & 2end{array}right] )
в. ( frac{1}{10}left[begin{array}{cc}2 & -2 \ 2 & 3end{array}right] )
c. ( frac{1}{10}left[begin{array}{cc}2 & 2 \ -2 & 3end{array}right] )
D. ( cdot frac{1}{10}left[begin{array}{cc}-2 & 2 \ 2 & 3end{array}right] )
12
33 If ( A ) is an ( m times n ) matrix and ( B ) is ( n times p )
matrix, then does ( A B ) exist? If yes, write
its order.
12
34 Find the matrix ( A ), such that ( left[begin{array}{l}4 \ 1 \ 3end{array}right] A= ) ( left[begin{array}{ccc}-4 & 8 & 4 \ -1 & 2 & 1 \ -3 & 6 & 3end{array}right] ) 12
35 If ( boldsymbol{A}=left[begin{array}{lll}1 & 1 & 1end{array}right], ) then ( A ) is a
A. Identity matrix
B. Null matrix
c. Diagonal matrix
D. Row matrix
12
36 If the order of ( mathbf{A} ) is ( mathbf{4} times mathbf{3}, ) the order of ( mathbf{B} )
is ( 4 times 5 ) and the order of ( C ) is ( 7 times 3 ), then
the order of ( left(mathbf{A}^{T} mathbf{B}right)^{T} mathbf{C}^{T} ) is
A. ( 4 times 5 )
в. ( 3 times 7 )
c. ( 4 x 3 )
D. ( 5 times 7 )
12
37 If ( boldsymbol{A}=left[begin{array}{ll}2 & 5 \ 4 & 4end{array}right], quad B=left[begin{array}{ll}0 & 5 \ 1 & 6end{array}right], ) find
( 5 boldsymbol{A}^{prime}+3 B^{prime} )
12
38 Consider ( A ) and ( B ) two square matrices
of same order. Select the correct
alternative.
( mathbf{A} cdot|A B| ) must be greater than ( |A| )
B. ( left[begin{array}{ll}1 & 1 \ 1 & 1end{array}right] ) is not unit matrix
C ( cdot|A+B| ) must be greater than ( |A| )
D. If ( A B=0 ), either ( A ) or ( B ) must be zero matrix
12
39 Find the adjoint of the matrix ( A= ) ( left[begin{array}{lll}1 & 4 & 3 \ 4 & 2 & 1 \ 3 & 2 & 2end{array}right] ) 12
40 If ( A=left|begin{array}{ccc}3 & -1 & 0 \ -1 & 2 & 3end{array}right| ) and ( B=left|begin{array}{ccc}-1 & 1 & 2 \ 0 & 2 & -1end{array}right| )
then find ( left(A B^{T}right)^{T} )
A ( cdotleft|begin{array}{cc}2 & -5 \ 1 & 6end{array}right| )
в. ( left|begin{array}{cc}3 & -4 \ 7 & -6end{array}right| )
с. ( left|begin{array}{cc}-4 & 9 \ -2 & 1end{array}right| )
D ( cdotleft|begin{array}{cc}-3 & 8 \ 4 & 2end{array}right| )
12
41 If matrices ( A ) and ( B ) anticommute then
( mathbf{A} cdot A B=B A )
B. ( A B=-B A )
c ( cdot(A B)=(B A)^{-1} )
D. None of these
12
42 Using elementary tansormations, find
the inverse of each of the matrices, if it
exists in ( left[begin{array}{ll}2 & 1 \ 1 & 1end{array}right] )
12
43 Assertion
If ( boldsymbol{F}(boldsymbol{alpha})=left[begin{array}{ccc}cos boldsymbol{alpha} & -sin boldsymbol{alpha} & boldsymbol{0} \ boldsymbol{s i n} boldsymbol{alpha} & boldsymbol{operatorname { c o s } boldsymbol { alpha }} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{0} & boldsymbol{1}end{array}right], ) then
( [boldsymbol{F}(boldsymbol{alpha})]^{-1}=boldsymbol{F}(-boldsymbol{alpha}) )
Reason For matrix ( G(beta)=left[begin{array}{ccc}cos beta & 0 & sin beta \ 0 & 1 & 0 \ -sin beta & 0 & cos betaend{array}right] )
we have ( [G(beta)]^{-1}=G(-beta) )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
44 If ( A ) is a ( 3 times 3 ) invertible matrix, then
what will be the value of ( k ) if
( operatorname{det}left(A^{-1}right)=(operatorname{det} A)^{k} )
12
45 Find ( x, y ) satisfying the matrix
equations ( left[begin{array}{ccc}boldsymbol{x}-boldsymbol{y} & boldsymbol{2} & -boldsymbol{2} \ boldsymbol{4} & boldsymbol{x} & boldsymbol{6}end{array}right]+left[begin{array}{ccc}boldsymbol{3} & -boldsymbol{2} & boldsymbol{2} \ boldsymbol{1} & boldsymbol{0} & boldsymbol{-} boldsymbol{1}end{array}right]= )
( left[begin{array}{ccc}mathbf{6} & mathbf{0} & mathbf{0} \ mathbf{5} & mathbf{2} boldsymbol{x}+boldsymbol{y} & mathbf{5}end{array}right] )
12
46 The number of ( A ) in ( T_{p} ) such that the trace of ( A ) is not divisible by ( p ) but ( operatorname{det}(A) ) divisible by p is ?[Note: The trace of matrix is the sum of its diagonal entries].
A ( cdot(p-1)left(p^{2}-p+1right) )
B . ( p^{3}-(p-1)^{2} )
c. ( (p-1)^{2} )
D. ( (p-1)left(p^{2}-2right) )
12
47 Assertion
The matrix ( boldsymbol{A}=left(begin{array}{ccc}mathbf{0} & boldsymbol{a} & boldsymbol{b} \ -boldsymbol{a} & mathbf{0} & boldsymbol{c} \ -boldsymbol{b} & -boldsymbol{c} & mathbf{0}end{array}right) ) is a
skew symmetric matrix.
Reason
A square matrix ( boldsymbol{A}=left(boldsymbol{a}_{boldsymbol{i} j}right) ) of order ( mathbf{m} ) is
said to be skew symmetric if ( boldsymbol{A}^{boldsymbol{T}}=-boldsymbol{A} )
A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
B. Both (A) &
(R) are individually true but (R) is not the correct (proper) explanation of (A).
C. (A)is true but (R) is false,
D. (A)is false but (R) is true.
12
48 If ( boldsymbol{A} ) is a square matrix such that ( boldsymbol{A}^{2}= )
( I, ) then find the simplified value of ( (A- )
( boldsymbol{I})^{3}+(boldsymbol{A}+boldsymbol{I})^{3}-mathbf{7} boldsymbol{A} )
12
49 Find the values of ( x ) and ( y, ) if ( left[begin{array}{cc}boldsymbol{x}+mathbf{1 0} & boldsymbol{y}^{2}+mathbf{2} boldsymbol{y} \ mathbf{0} & -mathbf{4} \ mathbf{3} boldsymbol{x}+mathbf{4} & mathbf{3} \ mathbf{0} & boldsymbol{y}^{mathbf{2}}-mathbf{5} boldsymbol{y}end{array}right]= ) 12
50 If ( boldsymbol{A}=left|begin{array}{cc}mathbf{5} & boldsymbol{x}-mathbf{2} \ mathbf{2} boldsymbol{x}+mathbf{3} & boldsymbol{x}+mathbf{1}end{array}right| ) is symmetric
( operatorname{then} x= )
A .4
B. 5
c. -5
D. –
12
51 ( operatorname{Let} boldsymbol{A}+mathbf{2} boldsymbol{B}=left[begin{array}{ccc}mathbf{1} & mathbf{2} & mathbf{0} \ mathbf{6} & -mathbf{3} & mathbf{3} \ -mathbf{5} & mathbf{3} & mathbf{1}end{array}right] ) and
( mathbf{2} boldsymbol{A}-boldsymbol{B}=left[begin{array}{ccc}mathbf{2} & mathbf{- 1} & mathbf{5} \ mathbf{2} & -mathbf{1} & mathbf{6} \ mathbf{0} & mathbf{1} & mathbf{2}end{array}right], ) then ( boldsymbol{t} boldsymbol{r}(boldsymbol{A}) )
( t r(B) ) has the value equal to
( A )
B.
( c cdot 2 )
D. none of these
12
52 If ( boldsymbol{A}=left(begin{array}{c}-121 \ 1 & 23end{array}right), B=(1) ) and ( C=(21) )
verify ( (A B) C=A(B C) )
12
53 Matrix A shows the weight of four boys and four girls in kg at the beginning of a diet programme to lose weight. Matrix B shows the corresponding weights after
the diet programme. ( boldsymbol{A}=left[begin{array}{llll}mathbf{3 5} & mathbf{4 0} & mathbf{2 8} & mathbf{4 5} \ mathbf{4 2} & mathbf{3 8} & mathbf{4 1} & mathbf{3 0}end{array}right] underset{mathbf{G i r l s}}{mathbf{B o y s}}, mathbf{B}= )
( left[begin{array}{cccc}mathbf{3 2} & mathbf{3 5} & mathbf{2 7} & mathbf{4 1} \ mathbf{4 0} & mathbf{3 0} & mathbf{3 4} & mathbf{2 7}end{array}right] begin{array}{c}boldsymbol{B} mathbf{o y s} \ text {Girls}end{array} )
Find the weight loss of the Boys and Girls.
12
54 Out of the following matrices, choose that matrix which is a scalar matrix.
( A cdotleft[begin{array}{ll}0 & 0 \ 0 & 0end{array}right] )
В. ( left[begin{array}{lll}0 & 0 & 0 \ 0 & 0 & 0end{array}right] )
c. ( left[begin{array}{ll}0 & 0 \ 0 & 0 \ 0 & 0end{array}right] )
D. ( left[begin{array}{l}0 \ 0 \ 0end{array}right] )
12
55 If ( A=operatorname{diag}left[d_{1}, d_{2}, d_{3}right] ) then ( a^{n} ) is equal to
A ( cdot operatorname{diag}left[d_{1}^{n-1}, d_{2}^{n-1}, d_{3}^{n-1}right] )
B. A
c. ( operatorname{diag}left[d_{1}^{n}, d_{2}^{n}, d_{3}^{n}right] )
D. none
12
56 Write the element ( a_{21} ) of the matrix
( boldsymbol{A}=left[boldsymbol{a}_{i j}right]_{2 times 2} ) whose elements ( boldsymbol{a}_{i j} ) are
( operatorname{given} ) by ( a_{a j}=e^{2 i x} cos j x )
12
57 [
operatorname{ftg}left[begin{array}{ccc}
3 & 2 & -1 \
2 & -2 & 0 \
1 & 3 & 1
end{array}right], Bleft[begin{array}{ccc}
-3 & -1 & 0 \
2 & 1 & 3 \
4 & -1 & 2
end{array}right]
]
and ( X=A+B ) then find ( X )
12
58 If ( mathbf{A} ) is a non-singular square matrix of order ( 3 times 3, ) find ( |a d j A| ) 12
59 fthe matrix ( boldsymbol{A}=left[begin{array}{lll}2 & 0 & 0 \ 0 & 2 & 0 \ 2 & 0 & 2end{array}right], ) then
( boldsymbol{A}^{n}=left[begin{array}{lll}boldsymbol{a} & boldsymbol{0} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{a} & boldsymbol{0} \ boldsymbol{b} & boldsymbol{0} & boldsymbol{a}end{array}right] . boldsymbol{n} in boldsymbol{N} ) where
A ( cdot a=2 n, b=2^{n} )
B . ( a=2^{n}, b=2 n )
c. ( a=2^{n}, b=n 2^{n-1} )
D. ( a=2^{n}, b=n 2^{n} )
12
60 Three roots of ( n ) are
A. 0,1,2
в. -1,1,3
( mathrm{c} .-2,2,3 )
D. -3,1,5
12
61 Find the value of ( x ) in ( left[begin{array}{cc}2 x-y & 5 \ 3 & yend{array}right]= )
( left[begin{array}{cc}mathbf{6} & mathbf{5} \ mathbf{3} & -mathbf{2}end{array}right] )
12
62 ( mathbf{f} boldsymbol{A}=left|begin{array}{cc}mathbf{2} & mathbf{0} \ mathbf{5} & -mathbf{3}end{array}right| boldsymbol{B}=left|begin{array}{cc}mathbf{- 2} & mathbf{1} \ mathbf{3} & mathbf{-} mathbf{1}end{array}right|, ) then
the find the trace of ( left(A B^{T}right)^{T} )
A . 10
B. 12
( c cdot 14 )
D. 16
12
63 ( left[begin{array}{ll}boldsymbol{x}+boldsymbol{y} & boldsymbol{2} \ boldsymbol{5}+boldsymbol{z} & boldsymbol{x} boldsymbol{y}end{array}right]=left[begin{array}{ll}boldsymbol{6} & boldsymbol{2} \ boldsymbol{5} & boldsymbol{8}end{array}right] )
Find ( boldsymbol{x} boldsymbol{y} boldsymbol{z} ? )
12
64 If ( boldsymbol{A}=left(begin{array}{c}cos boldsymbol{alpha} sin boldsymbol{alpha} \ -sin alpha cos boldsymbol{alpha}end{array}right), ) find ( boldsymbol{alpha} )
satisfying ( 0<alpha<frac{pi}{2} ) when ( A+A^{T}= )
( sqrt{2} I_{2} ; ) where ( A^{T} ) is transpose of ( A )
12
65 -b7
34.
If A=
and A adj A = A A”, then 5a +b is equal
[JEE M 2016]
to:
(a) 4
(c) 1
(b) 13
(d) 5
12
66 ( left.left[begin{array}{cc}mathbf{2 x}+boldsymbol{y} & boldsymbol{y} \ mathbf{1 – x} & mathbf{4 x}end{array}right]=mid begin{array}{cc}mathbf{1} & mathbf{- 1} \ mathbf{0} & mathbf{4}end{array}right], ) find the
values of ( boldsymbol{x}+boldsymbol{y} )
12
67 ( A_{n times n} ) and ( B_{n times n} ) are diagonal matrices
then ( A B=_{-dots} dots dots dots . . . ) matrix
This question has multiple correct options
A. square
B. diagonal
c. scalar
D. rectangular
12
68 ( left[begin{array}{lll}boldsymbol{x} & boldsymbol{4} & boldsymbol{1}end{array}right]left[begin{array}{ccc}boldsymbol{2} & boldsymbol{1} & boldsymbol{2} \ boldsymbol{1} & boldsymbol{0} & boldsymbol{2} \ boldsymbol{0} & boldsymbol{2} & boldsymbol{-} boldsymbol{4}end{array}right]left[begin{array}{c}boldsymbol{x} \ boldsymbol{4} \ -boldsymbol{1}end{array}right]=mathbf{0} )
then find ( x )
12
69 ( mathbf{f} boldsymbol{A}=left[begin{array}{ccc}-mathbf{1} & mathbf{2} & mathbf{0} \ -mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{0} & mathbf{1} & mathbf{0}end{array}right], ) Show that
( boldsymbol{A}^{2}=boldsymbol{A}^{-1} )
12
70 A skew-symmetric matrix ( M ) satisfies
the relation ( M^{2}+I=0, ) where ( I ) is the
unit matrix. Then, ( M M^{prime} ) is equal to
A . ( I )
B . ( 2 I )
( c .-I )
D. None of these
12
71 If ( boldsymbol{A}=left[begin{array}{lll}mathbf{6} & mathbf{1 0} & mathbf{1 0 0} \ mathbf{7} & mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{9} & mathbf{1 0}end{array}right], ) then
( operatorname{Tr}left(boldsymbol{A}^{T}right)= )
(Tr denotes trace of a matrix)
A . -17
B. 17
( c cdot-frac{1}{15} )
D. ( frac{1}{17} )
12
72 Identify the order of matrix ( left[begin{array}{ccc}1 & 0 & -4 \ 2 & -1 & 3end{array}right] ) 12
73 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{5} \ mathbf{6} & mathbf{7}end{array}right], ) then find ( boldsymbol{A}+boldsymbol{A}^{prime} ) 12
74 State whether the following statement
is true or false.
Enter 1 for true and 0 for false If ( boldsymbol{A}=left[begin{array}{ll}boldsymbol{a} & boldsymbol{b} \ boldsymbol{c} & boldsymbol{d}end{array}right] ) then the value of ( mathrm{f} ) and ( mathrm{g} )
satisfying the matrix equation ( A^{2}+ )
( boldsymbol{f} boldsymbol{A}+boldsymbol{g} boldsymbol{I}=boldsymbol{O} ) are equal to ( -boldsymbol{t}_{boldsymbol{r}}(boldsymbol{A}) ) and
determinant of A respectively. Given a b, ( c, d ) are non zero reals and ( I= ) ( left[begin{array}{ll}mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{1}end{array}right] ; boldsymbol{O}=left[begin{array}{ll}mathbf{0} & mathbf{0} \ mathbf{0} & mathbf{0}end{array}right] )
12
75 If ( mathbf{A}-mathbf{2 B}=left[begin{array}{cc}mathbf{1} & -mathbf{2} \ mathbf{3} & mathbf{0}end{array}right] ) and ( mathbf{2} mathbf{A}-mathbf{3} mathbf{B}= )
( left[begin{array}{cc}-mathbf{3} & mathbf{3} \ mathbf{1} & mathbf{- 1}end{array}right], ) then ( mathbf{B}= )
A ( cdotleft[begin{array}{cc}-5 & 7 \ 5 & 1end{array}right] )
B. ( left[begin{array}{cc}-5 & 7 \ -5 & -1end{array}right] )
c. ( left[begin{array}{cc}-5 & 7 \ 5 & -1end{array}right] )
D. ( left[begin{array}{cc}-5 & -7 \ -5 & -1end{array}right] )
12
76 Find the inverse of the following matrix, using elementary transformations:
( boldsymbol{A}=left[begin{array}{lll}mathbf{2} & mathbf{3} & mathbf{1} \ mathbf{2} & mathbf{4} & mathbf{1} \ mathbf{3} & mathbf{7} & mathbf{2}end{array}right] )
12
77 Number of real values of ( left|begin{array}{ccc}mathbf{3}-boldsymbol{x} & mathbf{2} & mathbf{2} \ mathbf{2} & mathbf{4}-boldsymbol{x} & mathbf{1} \ mathbf{- 2} & mathbf{- 4} & mathbf{- 1}-boldsymbol{x}end{array}right| ) is singular
then
( A cdot 1 )
B. 3
( c cdot 2 )
D. infinite
12
78 ff ( C=left[begin{array}{cc}3 & -6 \ 0 & 9end{array}right] ) find
¡) 2C
ii) ( frac{1}{3} C )
iii) –
12
79 Find the inverse of matrices by
elementary row transformation.
12
80 Let ( A ) be an invertible matrix then which
of the following is/are true
This question has multiple correct options
A ( cdotleft|A^{-1}right|=|A|^{-1} )
– ( left.^{-1}|A|^{-1}|=| Aright|^{-1} )
B. ( left(A^{2}right)^{-1}=left(A^{-1}right)^{2} )
C ( cdotleft(A^{T}right)^{-1}=left(A^{-1}right)^{T} )
D. none of these
12
81 If ( boldsymbol{A}=left(begin{array}{cc}mathbf{7} & mathbf{2} \ mathbf{1} & mathbf{3}end{array}right) ) and ( boldsymbol{A}+boldsymbol{B}= )
( left(begin{array}{cc}mathbf{- 1} & mathbf{0} \ mathbf{2} & mathbf{- 4}end{array}right) ) then matrix ( boldsymbol{B}=mathbf{?} )
( mathbf{A} cdotleft(begin{array}{cc}1 & 0 \ 1 & 1end{array}right) )
B. ( left(begin{array}{cc}6 & 2 \ 3 & -1end{array}right) )
c. ( left(begin{array}{cc}-8 & -2 \ 1 & -7end{array}right) )
D. ( left(begin{array}{cc}8 & 2 \ -1 & 7end{array}right) )
12
82 Find the inverse of the matrix
( left[begin{array}{ccc}1 & 2 & 1 \ -1 & 0 & 2 \ 2 & 1 & -3end{array}right] ) by elementary row
transformation. Hence solve the system
of equations ( boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{4},-boldsymbol{x}+ )
( mathbf{2} z=mathbf{0}, mathbf{2} boldsymbol{x}+boldsymbol{y}-mathbf{3} boldsymbol{z}=mathbf{0} )
12
83 ( operatorname{Let} boldsymbol{A}=left[begin{array}{c}-mathbf{1} \ mathbf{2} \ mathbf{3}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{lll}-mathbf{2} & -mathbf{1} & -mathbf{4}end{array}right] )
If trace of matrix ( A B ) is -12 , then the
value of ( k )
( A cdot 7 )
B.
( c cdot 2 )
D. none of these
12
84 ( operatorname{Let} boldsymbol{a}_{boldsymbol{k}}=boldsymbol{k}left(^{10} boldsymbol{C}_{boldsymbol{k}}right), boldsymbol{b}_{boldsymbol{k}}=(mathbf{1 0}-boldsymbol{k})left(^{10} boldsymbol{C}_{boldsymbol{k}}right) )
and ( boldsymbol{A}_{boldsymbol{k}}=left[begin{array}{ll}boldsymbol{a}_{boldsymbol{k}} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{b}_{boldsymbol{k}}end{array}right] )
If ( boldsymbol{A}=sum_{boldsymbol{k}=1}^{9} boldsymbol{A}_{boldsymbol{k}}=left[begin{array}{ll}boldsymbol{a} & mathbf{0} \ mathbf{0} & boldsymbol{b}end{array}right], ) find the value
of ( a+b )
12
85 ( boldsymbol{A}=left[begin{array}{lll}mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{4} & mathbf{5} & mathbf{6} \ mathbf{7} & mathbf{1} & mathbf{0}end{array}right], boldsymbol{B}=left[begin{array}{lll}mathbf{1} & mathbf{0} & mathbf{0} \ mathbf{0} & mathbf{3} & mathbf{0} \ mathbf{0} & mathbf{4} & mathbf{5}end{array}right] )
( operatorname{Tr}(A B)=lambda operatorname{Tr}(mathrm{A}) . operatorname{Tr}(mathrm{B}), ) then ( lambda= )
( A )
B. 0
( c cdot frac{6}{5} )
( frac{20}{27} )
12
86 ( A ) and ( B ) are two square matrices of
same order. If ( A B=B^{-1}, ) then ( A^{-1}= )
A. ( B A )
B . ( A^{2} )
( c cdot B^{2} )
D. ( B )
12
87 If ( P ) is a two-rowed matrix satisfying
( P^{T}=P^{-1}, ) then ( P ) can be
( mathbf{A} cdotleft[begin{array}{cc}cos theta & -sin theta \ -sin theta & cos thetaend{array}right] )
B. ( left[begin{array}{cc}cos theta & sin theta \ -sin theta & cos thetaend{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}-cos theta & sin theta \ sin theta & -cos thetaend{array}right] )
D. none of these
12
88 ff ( A=left[begin{array}{ccc}1 & 3 & 1 \ 2 & 1 & -1 \ 3 & 0 & 1end{array}right], ) then ( operatorname{rank}(A) ) is
equal to
( mathbf{A} cdot mathbf{4} )
B.
( c cdot 2 )
D. 3
12
89 Elements of a matrix ( A ) of order ( 10 times 10 )
are defined as ( a_{i j}=w^{i+j} ) (where ( w ) is
cube root of unity), then trace ( (A) ) of the
matrix is
( A cdot omega )
B.
( c cdot omega^{2} )
D.
12
90 ( mathbf{A}=left[begin{array}{ccc}mathbf{1} & mathbf{2} & mathbf{- 3} \ mathbf{5} & mathbf{0} & mathbf{2} \ mathbf{1} & mathbf{- 1} & mathbf{1}end{array}right] ) and ( mathbf{B}= )
( left[begin{array}{ccc}3 & -1 & 2 \ 4 & 2 & 5 \ 2 & 0 & 3end{array}right] )
Find the matrix ( C ) satisfying the relation
( A+2 C=B )
12
91 If ( left(begin{array}{c}3 x+7 \ y+1 & 2-3 xend{array}right)=left(begin{array}{cc}1 & y-2 \ 8 & 8end{array}right) ) then the values
of ( x ) and ( y ) respectively are
A. -2,7
B. ( -frac{1}{3}, 7 )
( c cdot-frac{1}{3},-frac{2}{3} )
D. 2,-7
12
92 Using elementary row transformation, find the inverse of ( left[begin{array}{ccc}2 & 0 & -1 \ 5 & 1 & 0 \ 0 & 1 & 3end{array}right] ) 12
93 If ( 2left[begin{array}{ll}3 & 4 \ 5 & xend{array}right]+left[begin{array}{ll}1 & y \ 0 & 1end{array}right]=left[begin{array}{cc}7 & 0 \ 10 & 5end{array}right], ) find
( (x-y) )
12
94 ff ( left[begin{array}{ll}boldsymbol{x} & mathbf{0} \ mathbf{1} & boldsymbol{y}end{array}right]+left[begin{array}{cc}-mathbf{2} & mathbf{1} \ mathbf{3} & mathbf{4}end{array}right]=left[begin{array}{ll}mathbf{3} & mathbf{5} \ mathbf{6} & mathbf{3}end{array}right]- )
( left[begin{array}{ll}2 & 4 \ 2 & 1end{array}right], ) then
A. ( x=-3, y=-2 )
в. ( x=3, y=-2 )
c. ( x=3, y=2 )
D. ( x=-3, y=2 )
12
95 Let ( boldsymbol{A}=left[begin{array}{c}-mathbf{1} \ mathbf{2} \ mathbf{3}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{lll}-mathbf{2} & -mathbf{1} & -mathbf{4}end{array}right] )
The skew symmetric part of ( A B ) is?
( mathbf{A} cdotleft(begin{array}{ccc}0 & 5 / 2 & 5 \ -5 / 2 & 0 & -5 / 2 \ -5 & 5 / 2 & 0end{array}right) )
В. ( left(begin{array}{ccc}0 & 5 & 10 \ -5 & 0 & 11 \ -10 & -11 & 0end{array}right) )
( mathbf{c} cdotleft(begin{array}{ccc}0 & 3 & 5 \ -3 & 0 & 11 \ -5 & -11 & 0end{array}right) )
D. None of these
12
96 If ( 2 A+3 B=left[begin{array}{ccc}2 & -1 & 4 \ 3 & 2 & 5end{array}right] ) and ( A+2 B= )
( left[begin{array}{ccc}5 & 0 & 3 \ 1 & 6 & 2end{array}right] ) Then ( B ) is
A ( cdotleft[begin{array}{ccc}8 & -1 & 2 \ -1 & 10 & -1end{array}right] )
B. ( left[begin{array}{ccc}8 & 1 & 2 \ -1 & 10 & -1end{array}right] )
c. ( left[begin{array}{ccc}8 & 1 & -2 \ -1 & 10 & -1end{array}right] )
D. ( left[begin{array}{lll}8 & 1 & 2 \ 1 & 10 & 1end{array}right] )
12
97 B is said to be a skew symmetric
matrix, if
A. ( B^{T}=B )
B. B = 0
( c cdot B=-B )
D. ( B^{T}=-B )
12
98 If ( boldsymbol{A}=left(begin{array}{l}23 \ 4 \ 5end{array}right), ) then find the transpose of ( boldsymbol{A} ) 12
99 If ( [boldsymbol{x} quad boldsymbol{y}]=left[begin{array}{ll}1 & 5end{array}right], ) then ( 2 boldsymbol{x}+mathbf{5} boldsymbol{y}= )
A . 26
B. 27
c. 29
D. None of these
12
100 Evaluate
( left[begin{array}{lll}mathbf{3} & mathbf{4} & mathbf{1}end{array}right]left[begin{array}{c}mathbf{3} \ -mathbf{1} \ mathbf{3}end{array}right] )
12
101 If ( boldsymbol{A}=left[begin{array}{ccc}1 & 2 & 2 \ 2 & 1 & -2 \ a & 2 & bend{array}right] ) is a matrix
satisfying the equation ( boldsymbol{A} boldsymbol{A}^{boldsymbol{T}}=mathbf{9} boldsymbol{I} )
where ( I ) is ( 3 times 3 ) identity matrix, then
the ordered pair ( (a, b) ) is equal:
A ( .(2,-11) )
в. (-2,1)
c. (2,1)
()
D. (2,-1)
12
102 ( mathbf{A}=left[begin{array}{cc}mathbf{1 1} & mathbf{1} \ mathbf{0} & mathbf{1 1}end{array}right], mathbf{B}=left[begin{array}{cc}mathbf{0} & -mathbf{2} \ -mathbf{3} & mathbf{4}end{array}right], mathbf{I}=left[begin{array}{cc}mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{1}end{array}right] )
Find ( boldsymbol{A}+mathbf{3} boldsymbol{B}+boldsymbol{4} boldsymbol{I} )
A ( cdotleft[begin{array}{cc}0 & 2 \ -3 & 0end{array}right] )
B ( cdotleft[begin{array}{cc}15 & 2 \ -13 & -16end{array}right] )
c. ( left[begin{array}{cc}15 & -5 \ -9 & 27end{array}right] )
D. ( left[begin{array}{cc}11 & 2 \ -4 & -5end{array}right] )
12
103 If ( boldsymbol{A}=left[boldsymbol{a}_{i j}right] ) and ( boldsymbol{a}_{i j}=boldsymbol{i}(boldsymbol{i}+boldsymbol{j}) ) then
trace of ( boldsymbol{A}= )
A ( cdot frac{n(n+1)(2 n+1)}{6} )
B. ( frac{n(n+1)(2 n+1)}{3} )
c. ( frac{n(n+1)}{2} )
D. ( frac{n^{2}(n+1)^{2}}{4} )
12
104 ( left[begin{array}{ll}boldsymbol{x}-boldsymbol{y} & boldsymbol{4} \ boldsymbol{z}+boldsymbol{6} & boldsymbol{x}+boldsymbol{y}end{array}right]=left[begin{array}{ll}boldsymbol{8} & boldsymbol{w} \ boldsymbol{0} & boldsymbol{6}end{array}right], ) write the
value of ( (boldsymbol{x}+boldsymbol{y}+boldsymbol{z}) )
12
105 Consider the following relation ( R ) on the
set of real square matrices of order 3
( boldsymbol{R}={(boldsymbol{A}, boldsymbol{B}), mid boldsymbol{A} boldsymbol{B}=boldsymbol{B} boldsymbol{A}} )
STATEMENT-1: Relation R is
equivalence.
STATEMENT-2: Relation R is symmetric.
A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-
B. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
C. STATEMENT -1 is True, STATEMENT-2 is False
D. STATEMENT-1 is False, STATEMENT-2 is True
12
106 ( boldsymbol{A}=left[begin{array}{ll}mathbf{5} & mathbf{7} \ mathbf{9} & mathbf{4}end{array}right] boldsymbol{B}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{5}end{array}right] ) find ( boldsymbol{A}-boldsymbol{B} ) 12
107 If ( boldsymbol{A}^{prime}=left[begin{array}{cc}-2 & 3 \ 1 & 2end{array}right] ) and ( boldsymbol{B}=left[begin{array}{cc}-1 & 0 \ 1 & 2end{array}right] )
then find ( (boldsymbol{A}+mathbf{2} boldsymbol{B})^{prime} )
12
108 If ( boldsymbol{A}=left[begin{array}{ccc}2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2end{array}right], ) verify that
( boldsymbol{A}^{3}-mathbf{6} boldsymbol{A}^{2}+mathbf{9} boldsymbol{A}-mathbf{4} boldsymbol{I}=mathbf{0} . ) Hence find
( boldsymbol{A}^{-1} )
12
109 ( operatorname{Let} boldsymbol{A}+mathbf{2} boldsymbol{B}=left[begin{array}{ccc}mathbf{1} & mathbf{2} & mathbf{0} \ mathbf{6} & -mathbf{3} & mathbf{3} \ mathbf{- 5} & mathbf{3} & mathbf{1}end{array}right] ) and
( mathbf{2} boldsymbol{A}-boldsymbol{B}=left[begin{array}{ccc}mathbf{2} & mathbf{- 1} & mathbf{5} \ mathbf{2} & -mathbf{1} & mathbf{6} \ mathbf{0} & mathbf{1} & mathbf{2}end{array}right] ) then Det
( operatorname{Tr}(A)-operatorname{Tr}(B)) ) has the value equal to
( A )
B.
( c )
( D )
12
110 If ( boldsymbol{A}=left[begin{array}{cccc}mathbf{5} & mathbf{6} & mathbf{- 2} & mathbf{3} \ mathbf{1} & mathbf{0} & mathbf{4} & mathbf{2}end{array}right] ) and ( boldsymbol{B}= )
( left[begin{array}{cccc}mathbf{3} & mathbf{- 1} & mathbf{4} & mathbf{7} \ mathbf{2} & mathbf{8} & mathbf{2} & mathbf{3}end{array}right], ) then find ( mathbf{A}+mathbf{B} )
12
111 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{4}end{array}right], ) then number of
elements in ( A ) are
A .4
B. 3
( c cdot 2 )
D. None of these
12
112 ( operatorname{Let} A=left[begin{array}{ccc}1 & -1 & 1 \ 2 & 1 & -3 \ 1 & 1 & 1end{array}right] ) and ( 10 B= )
( left[begin{array}{ccc}4 & 2 & 2 \ -5 & 0 & alpha \ 1 & -2 & 3end{array}right] . ) If ( B ) is the inverse of
matrix ( A ), then ( alpha ) is
( A )
в.
( c cdot 2 )
( D )
12
113 If ( boldsymbol{A}=left(begin{array}{ccc}4 & 1 & 2 \ 1 & -2 & 3 \ 0 & 3 & 2end{array}right), boldsymbol{B}=left(begin{array}{ccc}2 & 0 & 4 \ 6 & 2 & 8 \ 2 & 4 & 6end{array}right) ) and ( boldsymbol{C}= )
( left(begin{array}{ccc}1 & 2 & -3 \ 5 & 0 & 2 \ 1 & -1 & 1end{array}right), ) then
verify that ( boldsymbol{A}+(boldsymbol{B}+boldsymbol{C})=(boldsymbol{A}+boldsymbol{B})+ )
( boldsymbol{C} )
12
114 If
(i) ( A=left[begin{array}{cc}cos alpha & sin alpha \ -sin alpha & cos alphaend{array}right], ) then verify
that ( boldsymbol{A}^{prime} boldsymbol{A}=boldsymbol{I} )
(ii) ( A=left[begin{array}{cc}sin alpha & cos alpha \ -cos alpha & sin alphaend{array}right], ) then verify
that ( boldsymbol{A}^{prime} boldsymbol{A}=boldsymbol{I} )
12
115 If ( mathbf{3} boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{2} & mathbf{2} \ mathbf{2} & mathbf{1} & -mathbf{2} \ boldsymbol{x} & mathbf{2} & boldsymbol{y}end{array}right], ) and ( boldsymbol{A} boldsymbol{A}^{prime}=boldsymbol{I} )
then ( -boldsymbol{x}-boldsymbol{y}=ldots )
12
116 Using elementary tansormations, find
the inverse of each of the matrices, if it
exists in ( left[begin{array}{ll}2 & 3 \ 5 & 7end{array}right] )
12
117 For ( k=frac{1}{sqrt{50}}, ) find ( a, b, c ) such that ( boldsymbol{P P}^{T}=boldsymbol{I} ) where, ( boldsymbol{P}=left[begin{array}{ccc}frac{2}{3} & boldsymbol{3} boldsymbol{k} & boldsymbol{a} \ -frac{1}{3} & -boldsymbol{4} boldsymbol{k} & boldsymbol{b} \ frac{2}{3} & -boldsymbol{5} boldsymbol{k} & boldsymbol{c}end{array}right] )
A ( cdot a=pm frac{13}{15 sqrt{2}}, b=pm frac{16}{15 sqrt{2}}, c=frac{1}{3 sqrt{2}} )
в. ( a=pm frac{13}{15 sqrt{2}}, b=pm frac{16}{15 sqrt{2}}, c=pm frac{1}{15 sqrt{2}} )
c. ( a=pm frac{13}{15 sqrt{2}}, b=pm frac{16}{15 sqrt{2}}, c=pm frac{1}{3 sqrt{2}} )
D. None of these
12
118 Prove that ( (A B)(A B)^{-1}=1 ) 12
119 If matrix ( mathbf{A}=[boldsymbol{a} boldsymbol{i} boldsymbol{j}]_{3 times 2}, ) and ( boldsymbol{a} boldsymbol{i} boldsymbol{j}=(boldsymbol{3} boldsymbol{i}- )
( 2 j)^{2}, ) then find the matrix ( A )
12
120 Find the values of a and ( b, ) if ( A=B, ) where
[
begin{array}{c}
boldsymbol{A}=left[begin{array}{cc}
boldsymbol{a}+boldsymbol{4} & boldsymbol{3} boldsymbol{b} \
boldsymbol{8} & -boldsymbol{6}
end{array}right] text { and } boldsymbol{B}= \
{left[begin{array}{cc}
boldsymbol{2} boldsymbol{a}+boldsymbol{2} & boldsymbol{b}^{2}+boldsymbol{2} \
boldsymbol{8} & boldsymbol{b}^{2}-boldsymbol{5} boldsymbol{b}
end{array}right]}
end{array}
]
12
121 If ( A ) and ( B ) are symmetric matrices, then ( (A B-B A) ) is skew-symmetric.
A. True
B. False
12
122 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{2} & mathbf{1} \ -mathbf{1} & mathbf{2}end{array}right], boldsymbol{B}=left[begin{array}{cc}mathbf{1} & -mathbf{2} \ mathbf{2} & mathbf{1}end{array}right], boldsymbol{C}= )
( left[begin{array}{cc}1 & -3 \ 2 & 1end{array}right], ) then which of the following is
true
A ( . A+B=B+A ) and ( A+(B+C)=(A+B)+C )
B. ( A+B=B+A ) and ( A C=B C )
c. ( A+B=B+A ) and ( A B=B C )
D. ( A C=B C ) and ( A=B C )
12
123 If ( mathbf{A}+left|begin{array}{cc}mathbf{4} & mathbf{2} \ mathbf{1} & mathbf{3}end{array}right|=left|begin{array}{cc}mathbf{6} & mathbf{9} \ mathbf{1} & mathbf{4}end{array}right| ) then ( mathbf{A}= )
( mathbf{A} cdotleft|begin{array}{ll}2 & 7 \ 0 & 1end{array}right| )
( mathbf{B} cdotleft|begin{array}{ll}0 & 1 \ 2 & 7end{array}right| )
( mathbf{C} cdotleft|begin{array}{ll}1 & 0 \ 2 & 7end{array}right| )
( mathbf{D} cdotleft|begin{array}{ll}2 & 1 \ 0 & 7end{array}right| )
12
124 If ( A^{T}=left[begin{array}{cc}3 & 4 \ -1 & 2 \ 0 & 1end{array}right] ) and ( B= )
( left[begin{array}{ccc}-1 & 2 & 1 \ 1 & 2 & 3end{array}right], ) then find ( A^{T}-B^{T} )
12
125 If ( boldsymbol{A}=left[begin{array}{c}-2 \ 4 \ 6end{array}right] ) and ( B=[14-6] ) then
find ( boldsymbol{A B} )
12
126 Find the inverse of the matrix
( left[begin{array}{ccc}1 & 0 & 0 \ 3 & 3 & 0 \ 5 & 2 & 1end{array}right] )
12
127 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{1} & mathbf{1} \ -mathbf{1} & mathbf{1}end{array}right] ) and ( quad boldsymbol{n} boldsymbol{epsilon} boldsymbol{N}, ) then ( boldsymbol{A}^{n} ) is
equal to
( mathbf{A} cdot 2^{n} A )
B ( cdot 2^{n-1} A )
( c cdot n A )
D. None of these
12
128 ( fleft|begin{array}{ll}boldsymbol{x} & boldsymbol{y} \ mathbf{1} & boldsymbol{6}end{array}right|=left|begin{array}{ll}mathbf{1} & mathbf{8} \ mathbf{1} & mathbf{6}end{array}right| ) then ( mathbf{x}+2 mathbf{y}= )
( A cdot 9 )
B. 17
c. 10
( D )
12
129 21.
The number of 3 x 3 non-singular matrices, with four entries
as 1 and all other entries as 0, is
[2010]
(a) je
(b) 6
(c) at least 7
(d) less than 4
12
130 If ( a, b, c ) and ( d ) are real numbers such that and if ( boldsymbol{A}=left|begin{array}{cc}boldsymbol{a}+boldsymbol{i b} & boldsymbol{c}+boldsymbol{i d} \ -boldsymbol{c}+boldsymbol{i d} & boldsymbol{a}-boldsymbol{i b}end{array}right| ) then
( A^{-1}= )
( mathbf{A} cdotleft|begin{array}{ll}a+i b & -c-i d \ c-i d & a-i bend{array}right| )
B. ( left|begin{array}{ll}a-i b & c+i d \ -c+i d & a+i bend{array}right| )
c. ( left|begin{array}{cc}a-i b & -c-i d \ c-i d & a+i bend{array}right| )
D. ( left|begin{array}{ll}a+i b & c+i d \ c-i d & a-i bend{array}right| )
12
131 The total number of matrices formed
with the help of 6 different numbers are
( mathbf{A} cdot mathbf{6} )
в. ( 6 ! )
c. ( 2(6 !) )
D. ( 4(6 !) )
12
132 Find ( x, ) if the matrix ( left|begin{array}{ccc}-1 & 2 & 3 \ 2 & 5 & 6 \ 3 & x & 7end{array}right| ) is a
symmetric matrix.
12
133 12. If A and B are square matrices of size n x n such that
AC – B4 = (A – B)(A+B), then which of the following will
be always true?
[2006]
(a) A=B
(b) AB = BA
(c) either of A or B is a zero matrix
(d) either of A or B is identity matrix
12
134 Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] )
whose element ( a_{i j} ) is ( a_{i j}=frac{(i-j)^{2}}{2} )
12
135 Find the order of the matrix
( left[begin{array}{ccc}1 & 1 & 3 \ 5 & 2 & 6 \ -2 & -1 & -3end{array}right] )
12
136 What must be the matrix ( boldsymbol{X} ) if ( 2 boldsymbol{X}+ ) ( left[begin{array}{l}12 \ 34end{array}right]=left[begin{array}{l}38 \ 72end{array}right] ? )
A ( cdotleft[begin{array}{l}13 \ 2-1end{array}right] )
в. ( left[begin{array}{l}1-3 \ 2-1end{array}right] )
c. ( left[begin{array}{l}26 \ 4-2end{array}right] )
D. ( left[begin{array}{l}2-6 \ 4-2end{array}right] )
12
137 Let ( A ) be an invertible matrix. Which of
the following is not true?
A ( cdot A^{-1}=|A|^{-1} )
B. ( left(A^{2}right)^{-1}=left(A^{-1}right)^{2} )
c. ( left(A^{T}right)^{-1}=left(A^{-1}right)^{T} )
D. None of these
12
138 [
begin{array}{l}
boldsymbol{A}_{mathbf{1}}=left[boldsymbol{a}_{1}right] \
boldsymbol{A}_{2}=left[begin{array}{lll}
boldsymbol{a}_{2} & boldsymbol{a}_{3} \
boldsymbol{a}_{boldsymbol{4}} & boldsymbol{a}_{5}
end{array}right] \
boldsymbol{A}_{boldsymbol{3}}=left[begin{array}{lll}
boldsymbol{a}_{boldsymbol{6}} & boldsymbol{a}_{boldsymbol{7}} & boldsymbol{a}_{boldsymbol{8}} \
boldsymbol{a}_{boldsymbol{9}} & boldsymbol{a}_{boldsymbol{1 0}} & boldsymbol{a}_{11} \
boldsymbol{a}_{boldsymbol{1 2}} & boldsymbol{a}_{boldsymbol{1 3}} & boldsymbol{a}_{14}
end{array}right] ldots ldots boldsymbol{A}_{boldsymbol{n}}=[ldots .]
end{array}
]
where ( quad a_{r}=left[log _{2} rright]([.] ) denotes
greatest integer. Then trance of ( boldsymbol{A}_{mathbf{1 0}} )
12
139 Find ( boldsymbol{A}, ) if ( boldsymbol{A}+boldsymbol{B}=left[begin{array}{ll}mathbf{5} & mathbf{2} \ mathbf{0} & mathbf{9}end{array}right] ) and ( boldsymbol{A}- )
( boldsymbol{B}=left[begin{array}{cc}-mathbf{3} & -mathbf{6} \ mathbf{4} & -mathbf{1}end{array}right] )
12
140 If the order of the matrix is ( 1 times 2, ) then it
is a
A. Row matrix
B. Column matrix
c. Square matrix
D. None of these
12
141 ( boldsymbol{A}=left[begin{array}{ll}boldsymbol{a} & boldsymbol{b} \ boldsymbol{b} & boldsymbol{a}end{array}right] ) and ( boldsymbol{A}^{2}=left[begin{array}{ll}boldsymbol{alpha} & boldsymbol{beta} \ boldsymbol{beta} & boldsymbol{alpha}end{array}right] ) then
( mathbf{A} cdot alpha=a^{2}+b^{2}, beta=2 a b )
B . ( alpha=a^{2}+b^{2}, beta=a^{2}-b^{2} )
C ( cdot alpha=2 a b, beta=a^{2}+b^{2} )
D . ( alpha=a^{2}+b^{2}, beta=a b )
12
142 Using elementary row transformations,
find the inverse of the matrix
( left[begin{array}{ccc}1 & 2 & 3 \ 2 & 5 & 7 \ -2 & -4 & 5end{array}right] )
12
143 Suppose ( A ) and ( B ) are two ( 3 times 3 ) non
singular matrices such that ( (A B)^{k}= )
( boldsymbol{A}^{k} boldsymbol{B}^{k} )
for ( k=2008,2009,2010, ) then
This question has multiple correct options
( mathbf{A} cdot A B^{-1} A^{-1}=B^{-1} )
B. ( A^{-1} B^{-2} A=B^{2} )
( mathbf{c} . A B=B A )
D. ( A^{-2} B^{2} A^{2}=left(A^{-1} B Aright)^{2} )
12
144 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{1} \ mathbf{1} & mathbf{1}end{array}right], ) then ( boldsymbol{A}^{100} ) is equal to
( mathbf{A} cdot 2^{100} A )
в. ( 2^{99} ) А
( c cdot 100 A )
D . ( 299 A )
12
145 If ( A ) be any ( m times n ) matrix and both ( A B )
and BA are defined prove that B should
be ( boldsymbol{m} times boldsymbol{n} ) matrix
12
146 Suppose ( A ) is any ( 3 times 3 ) non-singular
matrix and ( (boldsymbol{A}-mathbf{3} boldsymbol{I})(boldsymbol{A}-mathbf{5} boldsymbol{I})=boldsymbol{O} )
where ( boldsymbol{I}=boldsymbol{I}_{3} ) and ( boldsymbol{O}=boldsymbol{O}_{3}, ) If ( boldsymbol{alpha} boldsymbol{A}+ )
( beta A^{-1}=4 I, ) then ( alpha+beta ) is equal to
A . 8
B. 7
c. 13
D. 12
12
147 If ( A ) and ( B ) are square matrices such
that ( B=-A^{-1} B A, ) then ( (A+B)^{2} ) is
equal to
( mathbf{A} cdot mathbf{0} )
B. ( A^{2}+B^{2} )
c. ( A^{2}+2 A B+B^{2} )
D. ( A+B )
12
148 ( mathbf{f} A=left[begin{array}{cc}cos boldsymbol{x} & -sin boldsymbol{x} \ sin boldsymbol{x} & cos boldsymbol{x}end{array}right], ) then find ( boldsymbol{A} boldsymbol{A}^{boldsymbol{T}} ) 12
149 f matrix ( boldsymbol{A}=left[begin{array}{cc}mathbf{2} & mathbf{- 2} \ mathbf{- 2} & mathbf{2}end{array}right] ) and ( boldsymbol{A}^{mathbf{2}}=boldsymbol{p} boldsymbol{A} )
then write the value of ( p )
12
150 Let ( A ) be a ( 3 times 3 ) matrix such that
[
boldsymbol{A} timesleft[begin{array}{lll}
mathbf{1} & mathbf{2} & mathbf{3} \
mathbf{0} & mathbf{2} & mathbf{3} \
mathbf{0} & mathbf{1} & mathbf{1}
end{array}right]=left[begin{array}{lll}
mathbf{0} & mathbf{0} & mathbf{1} \
mathbf{1} & mathbf{0} & mathbf{0} \
mathbf{0} & mathbf{1} & mathbf{0}
end{array}right]
]
Then ( boldsymbol{A}^{-1} ) is :
A.
[
left[begin{array}{lll}
3 & 2 & 1 \
3 & 2 & 0 \
1 & 1 & 0
end{array}right]
]
B.
[
left[begin{array}{lll}
0 & 1 & 3 \
0 & 2 & 3 \
1 & 1 & 1
end{array}right]
]
c.
[
left[begin{array}{lll}
3 & 1 & 2 \
3 & 0 & 2 \
1 & 0 & 1
end{array}right]
]
D.
[
left[begin{array}{lll}
1 & 2 & 3 \
0 & 1 & 1 \
0 & 2 & 3
end{array}right]
]
12
151 Inverse of the matrix ( left[begin{array}{cc}cos 2 theta & -sin 2 theta \ sin 2 theta & cos 2 thetaend{array}right] )
is.
( mathbf{A} cdotleft[begin{array}{cc}cos 2 theta & -sin 2 theta \ sin 2 theta & cos 2 thetaend{array}right] )
B. ( left[begin{array}{cc}cos 2 theta & sin 2 theta \ sin 2 theta & -cos 2 thetaend{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}cos 2 theta & sin 2 theta \ sin 2 theta & cos 2 thetaend{array}right] )
D. ( left[begin{array}{cc}cos 2 theta & sin 2 theta \ -sin 2 theta & cos 2 thetaend{array}right] )
12
152 If ( A ) is a square matrix then ( A-A^{prime} ) is a
A. diagonal matrix
B. skew symmetric matrix
c. symmetric matrix
D. None of these
12
153 If ( A ) is a skew-symmetric matrix of
order ( n ) and ( C ) is a column matrix of
order ( n times 1 ) then ( C^{T} A C ) is
A. a identity matrix of order ( n ).
B. a unit matrix of order
c. a zero matrix of order 1
D. none of these
12
154 If ( alpha ) and ( beta ) differ by an odd multiply on ( pi / 2 ) prove that the product of the two matrices given below is a null matrix 12
155 If ( boldsymbol{A}=left[begin{array}{l}mathbf{1} \ mathbf{2} \ mathbf{3}end{array}right], ) then find ( boldsymbol{A} boldsymbol{A}^{boldsymbol{T}} ) 12
156 If ( A ) is ( n ) squared matrix then ( A A^{prime} ) and
( A^{prime} A ) are symmetric.
12
157 ( A 2 times 2 ) matrix whose elements ( a_{i j} ) are
given by ( a_{i j}=i-j ) is
( mathbf{A} cdotleft[begin{array}{ll}0 & 1 \ 1 & 0end{array}right] )
B. ( left[begin{array}{cc}0 & -1 \ 1 & 0end{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}-1 & 0 \ 0 & 1end{array}right] )
D. ( left[begin{array}{cc}0 & 1 \ -1 & 0end{array}right] )
12
158 Choose the correct answer
A. Every scalar matrix is an identity matrix
B. Every identity matrix is a scalar matrix
C. Every diagonal matrix is an identity matrix
D. A square matrix whose each element is 1 is an identity matrix
12
159 If ( boldsymbol{A}=left[begin{array}{lll}boldsymbol{a}^{2} & boldsymbol{a b} & boldsymbol{a c} \ boldsymbol{a b} & boldsymbol{b}^{2} & boldsymbol{b c} \ boldsymbol{a c} & boldsymbol{b c} & boldsymbol{c}^{2}end{array}right] ) and ( boldsymbol{a}^{2}+boldsymbol{b}^{2}+ )
( c^{2}=1 ) then ( A^{2}= )
A ( .2 A )
в.
( c .3 A )
D. ( frac{1}{2} )
12
160 Which of the following property is not always true for metrics but in numbers?
A. ( A+B=0 )
B. AB = BA
( mathrm{c} cdot mathrm{AB}=0 )
D. None of the
12
161 If ( boldsymbol{A}=left[boldsymbol{a}_{i j}right] ) is a skew-symmetric matrix then write a value of ( sum_{i} sum_{j} a_{i j} ) 12
162 Write a ( 2 times 2 ) matrix which is both
symmetric and skew-symmetric.
12
163 7.
Let M be a 2 x 2 symmetric matrix with integer entries. Then
Mis invertible if
(JEE Adv. 2014)
(a) The first column of M is the transpose of the second
row of M
(b) The second row of Mis the transpose of the first column
of M
(c) Mis a diagonal matrix with non-zero entries in the main
diagonal
The product of entries in the main diagonal of M is not
the square of an integer
(d)
The
12
164 If ( A ) and ( B ) are two square matrices such
that ( B=-A^{-1} B A, ) then ( (A+B)^{2} ) is
equal to
( mathbf{A} cdot A^{2}+B^{2} )
в. ( O )
c. ( A^{2}+2 A B+B^{2} )
D. ( A+B )
12
165 If ( A ) and ( B ) are invertible matrices, which
one of the following statement is/are
correct
This question has multiple correct options
A ( . A d j(A)=|A| A^{-1} )
B. ( operatorname{det}left(A^{-1}right)=|operatorname{det}(A)|^{-1} )
c. ( (A+B)^{-1}=B^{-1}+A^{-1} )
D. ( (A B)^{-1}=B^{-1} A^{-1} )
12
166 If ( mathbf{A} ) and ( mathbf{B} ) are two square matrices
such that ( mathbf{B}=-mathbf{A}^{-1} mathbf{B} mathbf{A} ) then ( (mathbf{A}+ )
( mathbf{B})^{2}= )
A . 0
B. ( A^{2}+B^{2} )
c. ( A^{2}+2 A B+B^{2} )
D. ( A+B )
12
167 ( operatorname{Let} A=left[begin{array}{c}3 x^{2} \ 1 \ 6 xend{array}right], B=[a b c], ) and ( C= )
( left[begin{array}{ccc}(x+2)^{2} & 5 x^{2} & 2 x \ 5 x^{2} & 2 x & (x+2)^{2} \ 2 x & (x+2)^{2} & 5 x^{2}end{array}right] )
three given matrices, where ( a, b, c ) and ( x ) ( in mathrm{R} . ) Given that ( boldsymbol{t r}(boldsymbol{A B})=boldsymbol{t r}(boldsymbol{C}) boldsymbol{x} in boldsymbol{R} )
where ( operatorname{tr}(mathrm{A}) ) denotes trace of ( mathrm{A} ). If ( f(x)= ) ( a x^{2}+b x+c, ) then the value of ( f(1) ) is
12
168 ff ( left[begin{array}{cc}boldsymbol{x} & mathbf{2} \ mathbf{1 8} & boldsymbol{x}end{array}right]=left[begin{array}{cc}mathbf{6} & mathbf{2} \ mathbf{1 8} & mathbf{6}end{array}right] ) then ( mathbf{x}= )
( A cdot pm 6 )
B. 6
( c .-5 )
D.
12
169 If ( A^{prime} ) is the transpose of a square matrix
A, then
A ( cdot|A| neq mid A^{prime} )
B ( cdot|A|=left|A^{prime}right| )
c ( cdot|A|+left|A^{prime}right|=0 )
D cdot ( |A|=left|A^{prime}right| ) only when A is symmetric
12
170 Find the inverse of the following matrix by using elementary row transformation
( left[begin{array}{lll}0 & 1 & 2 \ 1 & 2 & 3 \ 3 & 1 & 1end{array}right] )
12
171 If ( boldsymbol{A}=left[begin{array}{c}mathbf{2} \ -mathbf{4} \ mathbf{1}end{array}right], boldsymbol{B}=left[begin{array}{lll}mathbf{5} & mathbf{3} & -mathbf{1}end{array}right] ) then
verify that ( (boldsymbol{A} boldsymbol{B})^{prime}=boldsymbol{B}^{prime} boldsymbol{A}^{prime} )
12
172 If ( boldsymbol{A}=left[begin{array}{ccc}boldsymbol{4} & mathbf{1} & boldsymbol{0} \ mathbf{1} & -mathbf{2} & mathbf{2}end{array}right], boldsymbol{B}= )
( left[begin{array}{ccc}mathbf{2} & mathbf{0} & -mathbf{1} \ mathbf{3} & mathbf{1} & mathbf{4}end{array}right], boldsymbol{C}=left[begin{array}{c}mathbf{1} \ mathbf{2} \ -mathbf{1}end{array}right] ) and
( (mathbf{3} boldsymbol{B}-mathbf{2} boldsymbol{A}) boldsymbol{C}+mathbf{2} boldsymbol{X}=mathbf{0} ) then ( boldsymbol{X}= )
A ( cdot frac{1}{2}left[begin{array}{c}3 \ 13end{array}right] )
в. ( frac{1}{2}left[begin{array}{c}3 \ -13end{array}right] )
c. ( frac{1}{2}left[begin{array}{c}-3 \ 13end{array}right] )
D. ( left[begin{array}{c}3 \ -13end{array}right] )
12
173 ( operatorname{Let} boldsymbol{A}=left[begin{array}{ccc}mathbf{2} & mathbf{0} & mathbf{7} \ mathbf{0} & mathbf{1} & mathbf{0} \ mathbf{1} & mathbf{- 2} & mathbf{1}end{array}right] ) and ( boldsymbol{B}= )
( left[begin{array}{ccc}-boldsymbol{x} & mathbf{1 4 x} & mathbf{7 x} \ mathbf{0} & mathbf{1} & mathbf{0} \ boldsymbol{x} & mathbf{- 4 x} & mathbf{- 2 x}end{array}right] ) are two matrices
such that ( A B=(A B)^{-1} ) and ( A B neq I )
(where ( I ) is an identity matrix of order
( 3 times 3) )
Find the value of
( boldsymbol{T} boldsymbol{r} cdotleft(boldsymbol{A} boldsymbol{B}+(boldsymbol{A} boldsymbol{B})^{2}+(boldsymbol{A} boldsymbol{B})^{3}+ldots+(boldsymbol{A} boldsymbol{B}right. )
where ( T r .(A) ) denotes the trace of
matrix ( boldsymbol{A} )
A. 98
в. 99
( c cdot 100 )
D. 10
12
174 ( fleft[begin{array}{ll}mathbf{4} & mathbf{3} \ boldsymbol{x} & mathbf{5}end{array}right]=left[begin{array}{ll}boldsymbol{y} & boldsymbol{z} \ mathbf{1} & mathbf{5}end{array}right] ) then find the value
of ( x, y ) and ( z )
12
175 ( mathbf{A B A}^{-1}=mathbf{X} ) then ( mathbf{B}^{mathbf{2}}= )
A ( cdot x^{2} )
B. AxA ( ^{-1} )
c. ( A x^{2} A^{-1} )
D. ( A^{-1} x^{2} A )
12
176 Assertion
If the matrices ( A, B,(A+B) ) are
nonsingular, then ( left[boldsymbol{A}(boldsymbol{A}+boldsymbol{B})^{-1} boldsymbol{B}right]^{-1}= )
( boldsymbol{B}^{-1}+boldsymbol{A}^{-1} )
Reason
( left[boldsymbol{A}(boldsymbol{A}+boldsymbol{B})^{-1} boldsymbol{B}right]^{-1}=left[boldsymbol{A}left(boldsymbol{A}^{-1}+right.right. )
( left.left.boldsymbol{B}^{-1}right) boldsymbol{B}right]^{-1}=left[left(boldsymbol{I}+boldsymbol{A} boldsymbol{B}^{-1}right) boldsymbol{B}right]^{-1}= )
( left[left(boldsymbol{B}+boldsymbol{A B}^{-1} boldsymbol{B}right)right]^{-1}=[(boldsymbol{B}+ )
( boldsymbol{A I})]^{-1}=[(boldsymbol{B}+mathbf{1})]^{-1}=boldsymbol{B}^{-1}+boldsymbol{A}^{-1} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
177 If ( A B=A ) and ( B A=B, ) then
This question has multiple correct options
( mathbf{A} cdot A^{2} B=A^{2} )
B. ( B^{2} A=B^{2} )
c. ( A B A=A )
D. ( B A B=B )
12
178 Find the rank of the matrix ( boldsymbol{A}= )
( left[begin{array}{ccc}1 & 5 & 9 \ 4 & 8 & 12 \ 7 & 11 & 15end{array}right] )
12
179 A square matrix A has 9 elements. What is the possible order of ( A ? )
( mathbf{A} cdot 1 times 9 )
B. ( 9 times 9 )
c. ( 3 times 3 )
D. ( 2 times 7 )
12
180 If ( A ) satisfies the equation ( x^{3}-5 x^{2}+ )
( 4 x+lambda=0, ) then ( A^{-1} ) exists if
A. ( lambda neq 1 )
B. ( lambda neq 2 )
c. ( lambda neq-1 )
D. ( lambda neq 0 )
12
181 If ( mathrm{A}=left[a_{i j}right]_{2 times 2} ) such that ( a_{i j}=i-j+3 )
then find ( boldsymbol{A} )
( mathbf{A} cdotleft[begin{array}{ll}2 & 3 \ 4 & 2end{array}right] )
в. ( left[begin{array}{ll}3 & 4 \ 2 & 3end{array}right] )
c. ( left[begin{array}{ll}4 & 2 \ 2 & 3end{array}right] )
D. ( left[begin{array}{ll}3 & 2 \ 4 & 3end{array}right] )
12
182 Construct a ( 2 times 3 ) matrix ( A=left[a_{i j}right] )
whose element ( a_{i j} ) is ( a_{i j}=2 i-j )
12
183 If ( boldsymbol{A}=left[begin{array}{ll}boldsymbol{a} & boldsymbol{x} \ boldsymbol{y} & boldsymbol{a}end{array}right] ) and if ( boldsymbol{x} boldsymbol{y}=mathbf{1}, ) then
( operatorname{det}left(A A^{T}right) ) is equal to:
( mathbf{A} cdot a^{2}-1 )
B ( cdotleft(a^{2}+1right)^{2} )
c. ( 1-a^{2} )
D. ( left(a^{2}-1right)^{2} )
E ( cdot(a-1)^{2} )
12
184 Multiply the given matrices:
[
left[begin{array}{cc}
1 & -2 \
2 & 3
end{array}right]left[begin{array}{ccc}
1 & 2 & 3 \
-3 & 2 & -1
end{array}right]
]
12
185 ( fleft(frac{2}{3} 1 frac{5}{3}right) quadleft[frac{2}{3} frac{2}{3} frac{4}{3}right] ) and ( B=left[frac{2}{5} frac{3}{5} frac{4}{5} frac{4}{5}right], ) then
compute ( 3 A-5 B )
12
186 If ( A ) and ( B ) are two non singular matrices of the same order such that
( B^{r}=I, ) for some positive integer ( r>1 )
( operatorname{then} boldsymbol{A}^{-1} boldsymbol{B}^{r-1} boldsymbol{A}-boldsymbol{A}^{-1} boldsymbol{B}^{-1} boldsymbol{A}= )
A . ( I )
B. ( 2 I )
( c cdot O )
D. ( -I )
12
187 ( boldsymbol{A}=left(begin{array}{ccc}1 & 0 & 1 \ 0 & 1 & 1 \ 0 & 1 & 0end{array}right) Rightarrow A^{2}-2 A= )
( A cdot A^{-1} )
B ( .-A^{-1} )
( c )
D. -1
12
188 Solve by matrix method ( boldsymbol{x}+mathbf{2} boldsymbol{y}+mathbf{3} boldsymbol{z}= )
( mathbf{2}, mathbf{2} boldsymbol{x}+mathbf{3} boldsymbol{y}+boldsymbol{z}=-mathbf{1}, boldsymbol{x}-boldsymbol{y}-boldsymbol{z}=-mathbf{2} )
12
189 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{4} & mathbf{2} \ mathbf{- 1} & mathbf{1}end{array}right], ) then ( (boldsymbol{A}-mathbf{2} boldsymbol{I})(boldsymbol{A}-mathbf{3} boldsymbol{I}) )
equals-
( mathbf{A} cdot mathbf{0} )
в.
c. ( I )
D. ( 5 I )
12
190 To find the inverse of the matrix ( boldsymbol{A}= )
( [121 ; 011 ; 311] ) by elementary
transformation method
12
191 ff ( boldsymbol{P}=left[begin{array}{ccc}2 & 3 & 1 \ 0 & -1 & 5end{array}right] ) and ( Q= )
( left[begin{array}{ccc}1 & 2 & -6 \ 0 & -1 & 3end{array}right], ) Evaluate ( 3 P-4 Q )
( mathbf{3} P-4 Q=left[begin{array}{lll}a & b & c \ d & e & fend{array}right], ) find sum of
( a, b, c, d, e, f )
12
192 A matrix consists of 30 elements. What
are the possible orders it can have?
12
193 ff ( left[begin{array}{cc}boldsymbol{alpha} & boldsymbol{beta} \ boldsymbol{gamma} & -boldsymbol{alpha}end{array}right] ) is to be the square root of ( mathbf{a} )
two -rowed unit matrix, then ( alpha, beta ) and ( gamma )
should satisfy the relation.
A. ( 1+alpha^{2}+beta gamma=0 )
B . ( 1-alpha^{2}-beta gamma=0 )
C ( cdot 1-alpha^{2}+beta gamma=0 )
D. ( 1+alpha^{2}-beta gamma=0 )
12
194 Let ( A, B, C, D ) be (not necessarily square) real matrices such that ( boldsymbol{A}^{boldsymbol{T}}= )
( boldsymbol{B} boldsymbol{C} boldsymbol{D} ; boldsymbol{B}^{T}=boldsymbol{C} boldsymbol{D} boldsymbol{A} ; boldsymbol{C}^{T}=boldsymbol{D} boldsymbol{A} boldsymbol{B} ) and
( D^{T}=A B C )
for the matrix ( S=A B C D ), consider the
two statements.
( boldsymbol{S}^{3}=boldsymbol{S} )
( | S^{2}=S^{4} )
A. II is true but not
B. I is true but not I
c. Both I and II are true
D. Both I and II are false.
12
195 ( mathrm{ff} mathrm{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{0} \ mathbf{4} & mathbf{1}end{array}right] ) and ( mathrm{B}=left[begin{array}{lll}mathbf{0} & mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{2} & mathbf{1} \ mathbf{2} & mathbf{3} & mathbf{1}end{array}right] ) find
BA. Can we find AB also?
12
196 If ( A ) and ( B ) are square matrices of same order and ( mathrm{B} ) is a skew-symmetric
matrix, show that ( A^{prime} B A ) is a skew
symmetric matrix.
12
197 If ( A ) and ( B ) are square matrices such that ( A B=I ) and ( B A=I, ) then ( B ) is
A. Unit matrix
B. Null matrix
c. Multiplicative inverse matrix of ( A )
D. ( -A )
12
198 f matrix ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{4} & mathbf{3}end{array}right] ) such that ( boldsymbol{A} boldsymbol{x}=boldsymbol{I} )
then ( boldsymbol{x}= )
begin{tabular}{r}
( mathbf{A} cdot frac{1}{5}left[begin{array}{rr}1 & 3 \
2 & -1end{array}right] ) \
hline
end{tabular}
в. 1
[
begin{array}{ll}overline{overline{5}} & {left[begin{array}{ll}4 & -1end{array}right]}end{array}
]
c. ( 1left[begin{array}{ll}-3 & 2 \ 5 & -1end{array}right] )
D. ( frac{1}{5}left[begin{array}{ll}-1 & 2 \ -1 & 4end{array}right] )
12
199 toppr LoGin Joln Now
Q Type your question
two classes of alphabets as in the two matrices given below. The columns and
rows of Matrix I are numbered from 0 to
4 and that of Matrix II are numbered
from 5 to ( 9 . ) A letter from these matrices
can be represented first by its row and
next by its column, e.g., ( ^{prime} boldsymbol{R}^{prime} ) can be
represented by 04,42 etc., and ( ^{prime} D^{prime} ) can
be represented by 57,76 etc. Similarly, you have to identify the set for the word ( mathrm{ROAD}^{prime} ) Matrix
1 begin{tabular}{cc|c|c|c|c}
& 0 & 1 & 2 & 3 & 4 \
hline 0 & ( F ) & ( O ) & ( M ) & ( S ) & ( R ) \
1 & ( S ) & ( R ) & ( F ) & ( O ) & ( M ) \
2 & ( O ) & ( M ) & ( S ) & ( R ) & ( F ) \
3 & ( R ) & ( F ) & ( O ) & ( M ) & ( S ) \
4 & ( M ) & ( S ) & ( R ) & ( F ) & ( O )
end{tabular}
Matrix I
begin{tabular}{cccc|cc}
& 5 & 6 & 7 & 8 & 9 \
5 & ( A ) & ( T ) & ( D ) & ( I ) & ( P ) \
6 & ( I ) & ( P ) & ( A ) & ( T ) & ( D ) \
7 & ( T ) & ( D ) & ( I ) & ( P ) & ( A ) \
8 & ( P ) & ( A ) & ( T ) & ( D ) & ( I ) \
9 & ( D ) & ( I ) & ( P ) & ( A ) & ( T )
end{tabular}
( A )
42,32,49,58
B. 23,32,98,99
( c )
11,13,67,69 13,67,69
00
( D )
12
200 Assertion
For a singular square matrix ( mathbf{A}, boldsymbol{A} boldsymbol{B}= )
( boldsymbol{A C} Rightarrow boldsymbol{B}=boldsymbol{C} )
Reason
If ( |A|=0, ) then ( A^{-1} ) does not exist.
A. Both Assertion and Reason are correct and Reason is
the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
201 If ( boldsymbol{A}=left[begin{array}{c}-2 \ 4 \ 5end{array}right] ) and ( B=left[begin{array}{lll}1 & 3 & -6end{array}right], ) then
verify that ( (A B)^{T}=B^{T} A^{T} )
12
202 IF ( A ) be a square matrix, then ( boldsymbol{A}+boldsymbol{A}^{boldsymbol{T}} )
is a symmetric matrix?
A. True
B. False
12
203 If ( A ) and ( B ) are symmetric matrices of the same order, write whether ( A B- )
( B A ) is symmetric or skew-symmetric or
neither of the two.
12
204 Let ( boldsymbol{P} ) be an ( boldsymbol{m} times boldsymbol{m} ) matrix such that
( boldsymbol{P}^{2}=boldsymbol{P} . ) Then ( (boldsymbol{I}+boldsymbol{P})^{n} ) equals.
( mathbf{A} cdot I+P )
B. ( I+n P )
c. ( I+2^{n} P )
D. ( I+left(2^{n}-1right) P )
12
205 ( fleft[begin{array}{cc}cos ^{2} alpha & cos alpha sin alpha \ cos alpha sin alpha & sin ^{2} alphaend{array}right] ) and ( B= )
( left[begin{array}{cc}cos ^{2} beta & cos beta sin beta \ cos beta sin beta & sin ^{2} betaend{array}right] ) are two
matrices such that the product ( A B ) is
null matirx, then ( alpha-beta ) is
( mathbf{A} cdot mathbf{0} )
B. Multiple of ( pi )
c. An odd number of ( frac{pi}{2} )
D. None of the above
12
206 If ( boldsymbol{A}^{prime}left[begin{array}{cc}-mathbf{2} & mathbf{3} \ mathbf{1} & mathbf{2}end{array}right], boldsymbol{B}=left[begin{array}{cc}-mathbf{1} & mathbf{0} \ mathbf{1} & mathbf{2}end{array}right] ) Find
( [boldsymbol{A}+mathbf{2} boldsymbol{B}] )
12
207 ( boldsymbol{A}=left[begin{array}{lll}2 & 3 & 1 \ 4 & 1 & 5 \ 3 & 9 & 7end{array}right] . ) Then the additive
inverse of ( boldsymbol{A} ) is:
( A )
[
left[begin{array}{ccc}
-2 & -3 & 1 \
4 & -1 & -5 \
-3 & 9 & -7
end{array}right]
]
B.
[
left[begin{array}{ccc}
-2 & -3 & -1 \
-4 & -1 & -5 \
-3 & -9 & -7
end{array}right]
]
( c )
[
left[begin{array}{ccc}
2 & -3 & -1 \
-4 & 1 & -5 \
-3 & -9 & 7
end{array}right]
]
D.
[
left[begin{array}{ccc}
-2 & -3 & -1 \
-4 & -1 & -5 \
-3 & 9 & -7
end{array}right]
]
12
208 If ( S=left[begin{array}{ll}6 & -8 \ 2 & 10end{array}right]=P+Q, ) where ( P ) is ( a )
symmetric & Q is a skew -symmetric matrix, then ( Q= )
A. ( left[begin{array}{cc}0 & 5 \ -5 & 0end{array}right] )
В. ( left[begin{array}{cc}0 & -5 \ 5 & 0end{array}right] )
C ( cdotleft[begin{array}{cc}0 & 8 \ -8 & 0end{array}right] )
D. ( left[begin{array}{cc}0 & 6 \ -6 & 0end{array}right] )
12
209 For any square matrix ( boldsymbol{A}, boldsymbol{A}+boldsymbol{A}^{T} ) is
A. unit matrix
B. symmetric matrix
c. skew symmetric matrix
D. zero matrix
12
210 ( mathbf{f} boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{- 2} & mathbf{0} \ mathbf{2} & mathbf{1} & mathbf{3} \ mathbf{0} & mathbf{- 2} & mathbf{1}end{array}right], ) then ( boldsymbol{A}^{-1}= ) 12
211 Express the matrix ( left[begin{array}{ccc}mathbf{3} & -mathbf{2} & mathbf{- 4} \ mathbf{3} & -mathbf{2} & -mathbf{5} \ -mathbf{1} & mathbf{1} & mathbf{2}end{array}right] ) as
the sum of a symmetric and skew-
symmetric matrix
12
212 If the matrix ( left[begin{array}{ccc}mathbf{0} & boldsymbol{a} & mathbf{3} \ mathbf{2} & boldsymbol{b} & mathbf{- 1} \ boldsymbol{c} & mathbf{1} & mathbf{0}end{array}right] ) is a skew
symmetric matrix, find ( a, b ) and ( c )
12
213 If ( boldsymbol{A}=operatorname{diag}(mathbf{2}-mathbf{5} mathbf{9}), boldsymbol{B}=operatorname{diag}(mathbf{1} mathbf{1}- )
4) and ( C=operatorname{diag}(-634), ) then find ( A )
( 2 B )
12
214 ( operatorname{Let} boldsymbol{A}=left[begin{array}{ll}mathbf{1} & frac{mathbf{3}}{2} \ mathbf{1} & mathbf{2}end{array}right], boldsymbol{B}= )
( left[begin{array}{cc}mathbf{4} & -mathbf{3} \ -mathbf{2} & mathbf{2}end{array}right] ) and ( boldsymbol{C}_{boldsymbol{r}}= )
( left[begin{array}{cc}r .3^{r} & 2^{r} \ 0 & (r-1) 3^{r}end{array}right] ) be 3 given matrices.
Compute the value of
( sum_{r=1}^{50} t r cdotleft((A B)^{r} C_{r}right) .(text { where } t r .(A) )
denotes trace of matrix ( mathbf{A} ) )
( mathbf{A} cdot 3left(49.3^{50}+1right) )
B ( cdot 3left(49.3^{49}+1right) )
C ( .3left(49.3^{48}+1right) )
D. None of these
12
215 Three school ( A, B ) and ( C ) organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold
hand made fans, mats and plates from
recycled material at a cost of Rs ( 25, ) Rs
100 and ( R s 50 ) each. The number of
articles sold are given below:
school ( begin{array}{lll}text { School } & text { School } & text { Sc } \ text { A } & text { B } & text { c }end{array} ) Articles
Hand40 25 fans
Mats 50 40
Plates 20 30 40
Find the fund collected by each school
separately by selling the above articles. Also, find the total funds collected for
the purpose.
12
216 ( mathbf{f} mathbf{A}=left[begin{array}{ccc}mathbf{1} & mathbf{0} & -mathbf{2} \ mathbf{2} & mathbf{- 3} & mathbf{4}end{array}right], ) then the matrix
( X ) for which ( 2 X+3 A=0 ) holds true is
( mathbf{A} cdotleft[begin{array}{ccc}-frac{3}{2} & 0 & -3 \ -3 & -frac{9}{2} & -6end{array}right] )
( mathbf{B} cdotleft[begin{array}{ccc}frac{3}{2} & 0 & -3 \ 3 & -frac{9}{2} & -6end{array}right] )
( mathbf{C} cdotleft[begin{array}{lll}frac{3}{2} & 0 & 3 \ 3 & frac{9}{2} & 6end{array}right] )
( ^{mathrm{D}} cdotleft[begin{array}{ccc}-frac{3}{2} & 0 & 3 \ -3 & frac{9}{2} & -6end{array}right] )
12
217 Prove that the matrix ( B^{prime} A B ) is
symmetric or skew symmetric
according as ( A ) is symmetric or skew symmetric.
12
218 If ( boldsymbol{A}^{-1}=left[begin{array}{cc}1 & -2 \ -2 & 2end{array}right], ) then what is
( operatorname{det}(A) ) equal to ( ? )
( A cdot 2 )
B. -2
c. ( 1 / 2 )
D. ( -1 / 2 )
12
219 If matrix ( A ) is an circulant matrix whose
elements of first row are ( mathbf{a}, mathbf{b}, mathbf{c} ) all ( >mathbf{0} )
such that abc ( =mathbf{1} )
and ( A^{T} A=I ) then ( a^{3}+b^{3}+c^{3} ) equals
( A cdot 0 )
B. 3
( c .1 )
D. 4
12
220 ( left[begin{array}{lll}0 & 0 & 0end{array}right] ) is
A. Identity matrix
B. diagonal matrix
c. scalar matrix
D. null matrix
12
221 Determine of ( U ) is
A . 13
B. 15
( c .3 )
D. 2
12
222 ( |f| A mid=47, ) then find ( left|A^{T}right| ) 12
223 Express the following matrices as the sum of a symmetric and a skew symmetric matrix :
( left[begin{array}{ccc}6 & -2 & 2 \ -2 & 3 & -1 \ 2 & -1 & 3end{array}right] )
12
224 The transpose of a rectangular matrix is
( mathbf{a} )
A. rectangular matrix
B. diagonal matrix
c. square matrix
D. scalar matrix
12
225 Which of the following is correct?
A. Determinant is a square matrix
B. Determinant is a number associated to a matrix
C. Determinant is a number associated to a square
matrix
D. None of these
12
226 ( boldsymbol{A}=left[boldsymbol{a}_{i j}right]_{m times n} ) is a square matrix , if
( mathbf{A} cdot mn )
c. ( m=n )
D. None of these
12
227 Let ( A ) and ( B ) are two matrices of same
order ( 3 times 3 ) given by ( A= ) ( left[begin{array}{ccc}1 & 3 & lambda+2 \ 2 & 4 & 6 \ 3 & 5 & 8end{array}right] B=left[begin{array}{ccc}3 & 2 & 4 \ 3 & 2 & 5 \ 2 & 1 & 4end{array}right] ) If ( lambda=4 )
( operatorname{then} frac{1}{6}{operatorname{tr}(A B)+operatorname{tr}(B A)} ) is equal to
( A cdot 42 )
B. 37
( c .35 )
D. None of thes
12
228 Find the symmetric and skew
symmetric parts of the matrix ( boldsymbol{A}=left[begin{array}{lll}1 & 2 & 4 \ 6 & 8 & 1 \ 3 & 5 & 7end{array}right] )
12
229 ( fleft(begin{array}{ccc}1 & -1 & 1 \ 2 & -1 & 0 \ 1 & 0 & 0end{array}right], ) then solve is ( A^{3}= )
I?
If correct state true else false.
A. True
B. False
12
230 Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] )
whose element ( a_{i j} ) is ( a_{i j}=frac{|-3 i+j|}{2} )
12
231 If ( A ) and ( B ) are square matrices of same order and ( A A^{T}=I ) then ( left(A^{T} B Aright)^{10} ) is
equal to
A. ( A B^{10} A^{T} )
B . ( A^{T} B^{10} A )
c. ( A^{10} B^{10}left(A^{T}right)^{10} )
D. ( 10 A^{T} B A )
12
232 ( boldsymbol{A}=left(begin{array}{cc}mathbf{3} & mathbf{2} \ -mathbf{1} & mathbf{4}end{array}right), boldsymbol{B}=left(begin{array}{cc}-mathbf{2} & mathbf{5} \ mathbf{6} & mathbf{7}end{array}right) ) and
( C=left(begin{array}{cc}1 & 1 \ -5 & 3end{array}right), ) then verify that
( boldsymbol{A}(boldsymbol{B}+boldsymbol{C})=boldsymbol{A} boldsymbol{B}+boldsymbol{A} boldsymbol{C} )
12
233 The maximum number of different
possible non-zero entries in a skew-
symmetric matrix of order ‘n’ is
A ( cdot frac{1}{2}left(n^{2}-nright) )
B ( cdot frac{1}{2}left(n^{2}+nright) )
( c cdot n^{2} )
D. ( left(n^{2}-nright) )
12
234 Find the value of ( x ) for which the matrix
product
[
left[begin{array}{ccc}
2 & 0 & 7 \
0 & 1 & 0 \
1 & -2 & 1
end{array}right]left[begin{array}{ccc}
-x & 14 x & 7 x \
0 & 1 & 0 \
x & -4 x & -2 x
end{array}right] text { equa }
]
to an identity matrix.
12
235 If ( A=[123] ) and ( B=left[begin{array}{l}1 \ 2 \ 3end{array}right], ) then find
( (A B)^{prime} )
12
236 Let ( A ) be a ( 3 times 3 ) matrix such that ( a_{11}= )
( a_{33}=2 ) and all the other ( a_{i j}=1 . ) Let
( boldsymbol{A}^{-1}=boldsymbol{x} boldsymbol{A}^{2}+boldsymbol{y} boldsymbol{A}-boldsymbol{z} boldsymbol{I}, ) then find the
value of ( (x+y+z) ) where ( I ) is a unit of
matrix of order 3
A . -9
B. 9
c. 1
D. –
12
237 Assertion
( boldsymbol{operatorname { T r }}(boldsymbol{A})=mathbf{0} )
Reason
( |A|=0 )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
238 For matrices ( A ) and ( B ; ) if ( A B=4 I, ) then
( A^{-1} ) is ( = )
A ( .4 B )
в. ( 4 B^{-1} )
c. ( frac{1}{4} )
D. ( frac{1}{4} B^{-1} )
12
239 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{4}end{array}right], ) such that ( boldsymbol{A} boldsymbol{X}=boldsymbol{I}, ) then
find ( boldsymbol{X} )
12
240 Find matrix ( boldsymbol{X} ) so that ( boldsymbol{X}left[begin{array}{lll}mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{4} & mathbf{5} & mathbf{6}end{array}right]= )
( left[begin{array}{ccc}-mathbf{7} & -mathbf{8} & -mathbf{9} \ mathbf{2} & mathbf{4} & mathbf{6}end{array}right] )
12
241 If ( A ) is a matrix of order ( 2 times 3, B ) is a
matrix of order ( 3 times 5 ), then what in the
order of matrix ( (A B) ) or ( (A B)^{t} )
12
242 If ( mathbf{A} ) is non-singular matrix such that ( A^{2}=A^{-1} ) then ( a d j A= )
A.
B. ( A^{-1} )
c. ( A^{3} )
D. ( left(mathrm{A}^{-1}right)^{2} )
12
243 Let ( A ) and ( B ) be matrices of orders ( 3 times 2 )
and ( 2 times 4 ) respectively. Write the order
of matrix ( boldsymbol{A} boldsymbol{B} )
12
244 If ( A=operatorname{diag}(1,-1,2) ) and ( B= )
( operatorname{diag}(2,3,-1) operatorname{then} 3 A+4 B= )
( operatorname{diag}(a, b, c) . ) Then ( a-b-c= )
12
245 If ( boldsymbol{A}_{1}, boldsymbol{A}_{3}, dots dots A_{2 n-1} ) are ( boldsymbol{n} ) skew
symmetric matrices of same order, then ( boldsymbol{B}=sum_{r=1}^{n}(2 r-1)left(A_{2 r-1}right)^{2 r-1} ) will be
A. symmetric
B. skew-symmetricç
c. neither symmetric nor skew-symmetric
D. data not adequate
12
246 ( mathbf{f} boldsymbol{A}=left[begin{array}{ccc}mathbf{0} & mathbf{- 1} & mathbf{2} \ mathbf{1} & mathbf{0} & mathbf{3} \ -mathbf{2} & mathbf{- 3} & mathbf{0}end{array}right], ) then ( boldsymbol{A}+mathbf{2} boldsymbol{A}^{prime} )
equals
A. ( A )
B . ( A^{prime} )
( c cdot-A )
D. ( 2 A )
12
247 f ( A ) and ( B ) are two matrices such that
( A B=B ) and ( B A=A ) and
( left(A^{2}+B^{2}right)=lambda(A+B) . ) Considering
( boldsymbol{f}(boldsymbol{x})=|[sin boldsymbol{x}]+[cos boldsymbol{x}]| ; ) where [] is
greatest integer function find ( boldsymbol{f}(mathbf{4} boldsymbol{lambda}) )
12
248 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{3} & mathbf{5} \ mathbf{7} & mathbf{- 9}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{cc}mathbf{6} & mathbf{- 4} \ mathbf{2} & mathbf{3}end{array}right], ) find
( (4 A-3 B) )
12
249 In a skew-symmetric matrix, the diagonal elements are all
A. one
B. zero
c. different from each other
D. non-zero
12
250 ff ( boldsymbol{I}=left[begin{array}{ll}mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{1}end{array}right] . boldsymbol{J}= )
( left[begin{array}{cc}mathbf{0} & mathbf{1} \ -mathbf{1} & mathbf{0}end{array}right] ) and ( quad boldsymbol{B}= )
( left[begin{array}{cc}cos theta & sin theta \ -sin theta & cos thetaend{array}right], quad ) then ( quad B= )
A ( . I cos theta+J sin theta )
B . ( operatorname{Isin} theta+J cos theta )
c. ( operatorname{Icos} theta-J sin theta )
D. ( -I cos theta-J sin theta )
12
251 ( operatorname{Matrix} boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{3} & mathbf{3} \ mathbf{2} & mathbf{4} & mathbf{1 0} \ mathbf{3} & mathbf{8} & mathbf{4}end{array}right] ) is similar to
( A )
[
left[begin{array}{lll}
1 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 1
end{array}right]
]
в.
[
left[begin{array}{ccc}
1 & 0 & 9 \
0 & 1 & -2 \
0 & 0 & -7
end{array}right]
]
( c )
[
left[begin{array}{lll}
1 & 0 & 5 \
0 & 1 & 2 \
0 & 4 & 1
end{array}right]
]
D. Both A and B
12
252 If ( boldsymbol{A}=left[begin{array}{cc}-boldsymbol{i} & mathbf{0} \ mathbf{0} & boldsymbol{i}end{array}right], ) then ( boldsymbol{A}^{prime} boldsymbol{A} ) is equal to
A . ( I )
B. ( -i A )
( c .-I )
D. ( i A )
12
253 Find the inverse of the following ( operatorname{matrices} boldsymbol{A}=left(begin{array}{ccc}2 & mathbf{0} & mathbf{1} \ mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{3} & mathbf{1} & mathbf{1}end{array}right) ) 12
254 Match the expression/statements on the left with the expression on the right. 12
255 ( mathrm{f} mathrm{A}=left[begin{array}{cc}1 & 3 \ 3 & 2 \ 2 & 5end{array}right], mathrm{B}=left[begin{array}{cc}-1 & -2 \ 0 & 5 \ 3 & 1end{array}right] ).Find the
matrices D such that ( boldsymbol{A}+boldsymbol{B}-boldsymbol{D}=boldsymbol{O} )
e. zero matrices
12
256 Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
(i) ( left[begin{array}{cc}mathbf{3} & mathbf{5} \ mathbf{1} & -mathbf{1}end{array}right] )
(ii) ( left[begin{array}{ccc}6 & -2 & 2 \ -2 & 3 & -1 \ 2 & -1 & 3end{array}right] )
(iii) ( left[begin{array}{ccc}mathbf{3} & mathbf{3} & -mathbf{1} \ -mathbf{2}-mathbf{2} & mathbf{1} \ -mathbf{4}-mathbf{5} & mathbf{2}end{array}right] )
(iv) ( left[begin{array}{cc}mathbf{1} & mathbf{5} \ -mathbf{1} & mathbf{2}end{array}right] )
12
257 ( operatorname{Let} A=left[begin{array}{ccc}1 & -1 & -1 \ 2 & 1 & -3 \ 1 & 1 & 1end{array}right] ) and ( 10 B= )
( left[begin{array}{ccc}mathbf{4} & mathbf{2} & mathbf{2} \ -mathbf{5} & mathbf{0} & boldsymbol{alpha} \ mathbf{1} & mathbf{- 2} & mathbf{3}end{array}right], ) if ( boldsymbol{B} ) is the inverse of
matrix ( A ), then ( alpha ) is
( A )
B.
( c cdot 2 )
( D )
12
258 If ( boldsymbol{A}=left[begin{array}{ccc}2 & 4 & -1 \ -1 & 0 & 2end{array}right], B=left[begin{array}{cc}3 & 4 \ -1 & 2 \ 2 & 1end{array}right] )
find ( (A B)^{T} )
12
259 If ( A ) and ( B ) are two matrices of same
order, then ( A+B ) is equal to
( mathbf{A} cdot B+A )
в. ( B A )
c. ( (A+B) T )
D. ( A-B )
12
260 If ( A ) and ( B ) are symmetric matrices of the same order and ( X=A B+B A ) and
( boldsymbol{Y}=boldsymbol{A} boldsymbol{B}-boldsymbol{B} boldsymbol{A}, ) then ( boldsymbol{X} boldsymbol{Y}^{boldsymbol{T}} ) is equal to
A . ( X Y )
в. ( Y X )
( mathrm{c} cdot-Y X )
D. none of these
12
261 ( mathrm{f}[mathbf{2} mathbf{1} mathbf{3}]left[begin{array}{ccc}-mathbf{1} & mathbf{0} & mathbf{1} \ -mathbf{1} & mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{1} & mathbf{1}end{array}right]left[begin{array}{l}mathbf{1} \ mathbf{0} \ mathbf{1}end{array}right]=boldsymbol{A}, ) then
write the order of matrix ( boldsymbol{A} )
12
262 In a upper triangular matrix ( n times n )
minimum number of zeroes is
A ( cdot frac{n(n-1)}{2} )
в. ( frac{n(n+1)}{2} )
c. ( frac{2 n(n-1)}{2} )
D. None of these
12
263 A fruit vendor sells fruits from his shop. Selling prices of Apple, Mango and Orange are Rs. ( 20, ) Rs. 10 and Rs. 5 each
respectively. The sales in three days are given below
( begin{array}{llll}text { Day } & text { Apples } & text { Mangoes } & text { Oranges } \ 1 & 50 & 60 & 30 \ 2 & 40 & 70 & 20 \ 3 & 60 & 40 & 10end{array} )
Write the matrix indicating the total amount collected on each day and
hence find the total amount collected
from selling of all three fruits combined.
12
264 If ( boldsymbol{A}=left[begin{array}{ll}boldsymbol{a} & boldsymbol{b}end{array}right], boldsymbol{B}=left[begin{array}{ll}-boldsymbol{b} & -boldsymbol{a}end{array}right] ) and ( boldsymbol{C}= )
( left[begin{array}{c}boldsymbol{a} \ -boldsymbol{a}end{array}right], ) then the correct statement is
A. ( A=-B )
В. ( A+B=A-B )
c. ( A C=B C )
D. ( C A=C B )
12
265 ( mathrm{f}left[begin{array}{l}mathbf{4} \ mathbf{1} \ mathbf{3}end{array}right] boldsymbol{A}=left[begin{array}{lll}-mathbf{4} & mathbf{8} & mathbf{4} \ -mathbf{1} & mathbf{2} & mathbf{1} \ -mathbf{3} & mathbf{6} & mathbf{3}end{array}right], ) if ( boldsymbol{A}= )
( boldsymbol{a} quad boldsymbol{b} quad boldsymbol{c}] )
Find ( a+b+c )
12
266 If ( boldsymbol{A}=left[boldsymbol{a}_{i j}right] ) is a scalar matrix of order
( boldsymbol{n} times boldsymbol{n} ) such that ( boldsymbol{a}_{boldsymbol{i} i}=boldsymbol{k} ) for all ( boldsymbol{i}, ) then
trace of ( boldsymbol{A} ) is equal to
( mathbf{A} cdot n k )
( mathbf{B} cdot n+k )
c. ( frac{n}{k} )
D. None of these
12
267 If ( boldsymbol{P}=left[begin{array}{ll}sqrt{mathbf{3}} / mathbf{2} & mathbf{1} / mathbf{2} \ -mathbf{1} / mathbf{2} & sqrt{mathbf{3}} / mathbf{2}end{array}right], boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{1} \ mathbf{0} & mathbf{1}end{array}right] )
( boldsymbol{Q}=boldsymbol{P} boldsymbol{A} boldsymbol{P}^{prime}, operatorname{then} boldsymbol{P}^{prime} boldsymbol{Q}^{2005} boldsymbol{P} ) is
A. ( left[begin{array}{cc}1 & 1 \ 2005 & 1end{array}right] )
B. ( left[begin{array}{cc}1 & 2005 \ 0 & 1end{array}right] )
c. ( left[begin{array}{cc}1 & 0 \ 0 & 1end{array}right] )
D. ( left[begin{array}{cc}1 & 2005 \ 2005 & 1end{array}right] )
12
268 If a matrix has 13 elements, then the
possible dimensions (orders) of the matrix are
( mathbf{A} cdot 1 times 13 ) or ( 13 times 1 )
B. ( 1 times 26 ) or ( 26 times 1 )
c. ( 2 times 13 ) or ( 13 times 2 )
D. ( 13 times 13 )
12
269 The matrix ( left[begin{array}{ll}0 & 1 \ 1 & 0end{array}right] ) is the matrix reflection in the line
( mathbf{A} cdot x=1 )
B . ( x+y=1 )
c. ( y=1 )
D. ( x=y )
12
270 If ( C ) is skew-symmetric matrix of order ( n )
and ( X ) in ( n times 1 ) column matrix, then ( X^{T} )
( mathrm{CX} ) is
This question has multiple correct options
A. singular
B. non-singular
c. invertible
D. non-invertible
12
271 Assertion ( operatorname{As} A=left[begin{array}{lll}2 & 1 & 1 \ 0 & 1 & 1 \ 1 & 1 & 2end{array}right] ) satisfies the
equation ( x^{3}-5 x^{2}+7 x-3=0 )
therefore ( A ) is invertible.
Reason
If a square matrix ( A ) satisfies the
equation ( a_{0} x^{n}+a_{1} x^{n-1}+ldots a_{n-1} x+ )
( a_{n}=0, ) and ( a_{n} neq 0, ) then ( A ) is invertible.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
272 If ( P=left[begin{array}{cc}frac{sqrt{3}}{2} & frac{1}{2} \ -frac{1}{2} & frac{sqrt{3}}{2}end{array}right], A=left[begin{array}{cc}1 & 1 \ 0 & 1end{array}right] ) and
( Q=P A P^{prime}, ) then ( P^{prime} Q^{2015} P ) is:
A. ( A=left[begin{array}{cc}0 & 2015 \ 0 & 0end{array}right] )
B. ( A=left[begin{array}{cc}2015 & 0 \ 1 & 2015end{array}right] )
c. ( A=left[begin{array}{cc}1 & 2015 \ 0 & 1end{array}right] )
D. ( A=left[begin{array}{cc}2015 & 1 \ 0 & 2015end{array}right] )
12
273 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{4}end{array}right], ) find ( boldsymbol{A}^{-1} ) by elementary
transformation
12
274 If the value of a third order determinant
is ( 11, ) then the value of the determinant
of ( A^{-1}= )
A . 1
B. 121
( c cdot 1 / 11 )
D. ( 1 / 121 )
12
275 The transpose of a column matrix is
A. zero matrix
B. diagonal matrix
c. column matrix
D. row matrix
12
276 Using elementary transformations find
the inverse of
( left[begin{array}{ccc}3 & 2 & 1 \ 2 & 4 & 3 \ 2 & -1 & 2end{array}right] )
12
277 If ( A ) is a ( 3 times 3 ) invertible matrix, then
what will be the value of ( k ) if
( operatorname{det}left(A^{-1}right)=(operatorname{det} A)^{k} )
12
278 If ( boldsymbol{A}=frac{mathbf{1}}{sqrt{mathbf{3}}}left[begin{array}{cc}mathbf{1} & boldsymbol{i}+mathbf{1} \ boldsymbol{i}-mathbf{1} & mathbf{1}end{array}right], ) then ( boldsymbol{A}left(boldsymbol{A}^{boldsymbol{T}}right) )
equals
( mathbf{A} cdot mathbf{0} )
B. ( I )
( c .-I )
D. ( 2 I )
12
279 If ( A^{2}+2 A+10=0 )
( A=left[begin{array}{lll}1 & 12 & 15 \ 2 & 10 & 4 \ 3 & 9 & 5end{array}right] ) find ( A^{-1} )
12
280 Write ( boldsymbol{A}=left[begin{array}{cc}mathbf{3} & mathbf{5} \ mathbf{1} & mathbf{-} mathbf{1}end{array}right] ) as the sum of ( mathbf{a} )
symmetric and a skew-symmetric
matrix.
12
281 ( a^{-1}+b^{-1}+c^{-1}=0 ) such that
( left|begin{array}{ccc}1+a & 1 & 1 \ 1 & 1+b & 1 \ 1 & 1 & 1+cend{array}right|=triangle ) then the
value of ( triangle ) is
( mathbf{A} cdot mathbf{0} )
B. abco
( mathrm{c} .-a b c )
D. None of these
12
282 ( boldsymbol{A B}=mathbf{0} ) where
( boldsymbol{A}=left[begin{array}{cc}cos ^{2} boldsymbol{theta} & cos boldsymbol{theta} sin boldsymbol{theta} \ cos boldsymbol{theta} sin boldsymbol{theta} & sin ^{2} boldsymbol{theta}end{array}right] )
( boldsymbol{B}=left[begin{array}{cc}cos ^{2} boldsymbol{phi} & cos phi sin phi \ cos phi sin phi & sin ^{2} phiend{array}right] )
then find ( |boldsymbol{theta}-boldsymbol{phi}|=? )
12
283 Using elementary tansormations, find
the inverse of each of the matrices, if it
exists in ( left[begin{array}{ll}2 & 5 \ 1 & 3end{array}right] )
12
284 ( f_{f} A=left[begin{array}{lll}0 & 1 & 2 \ 2 & 3 & 4 \ 4 & 5 & 6end{array}right], B=left[begin{array}{lll}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1end{array}right] )
Find ( 3 A-4 B )
12
285 Express the matrices as the sum of
systemmetric ( & ) a skew- symmetric
[
text { matrices }left[begin{array}{ccc}
mathbf{6} & -mathbf{2} & mathbf{2} \
-mathbf{2} & mathbf{3} & -mathbf{1} \
mathbf{2} & -mathbf{1} & mathbf{3}
end{array}right]
]
12
286 ( A ) is of order ( m times n ) and ( B ) is of order ( p times )
( q, ) addition of ( A ) and ( B ) is possible only if
A ( . m=p )
B . ( n=q )
c. ( n=p )
D. ( m=p, n=q )
12
287 If ( boldsymbol{A}= )
( frac{1}{pi}left[begin{array}{cc}sin ^{-1}(pi x) & tan ^{-1}left(frac{pi}{pi}right) \ sin ^{-1}left(frac{x}{pi}right) & cot ^{-1}(pi x)end{array}right], B= )
( frac{1}{pi}left[begin{array}{cc}-cos ^{-1}(pi x) & tan ^{-1}left(frac{x}{pi}right) \ sin ^{-1}left(frac{x}{pi}right) & -tan ^{-1}(pi x)end{array}right], ) then
( A-B ) is equal to
( A )
B.
( c .2 )
( D cdot underline{1} )
12
288 Suppose ( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{- 1} & mathbf{5} \ mathbf{4} & mathbf{3} & mathbf{7}end{array}right] ) find ( mathbf{4} boldsymbol{A} ) 12
289 ( left[begin{array}{cc}2 & 1 \ 1 & 0 \ -3 & 4end{array}right]+left[begin{array}{cc}-1 & -8 \ 1 & -2 \ 9 & 22end{array}right] ) 12
290 If ( boldsymbol{A}=left(begin{array}{rr}mathbf{4} & -mathbf{2} \ mathbf{5} & -mathbf{9}end{array}right) ) and ( boldsymbol{B}=left(begin{array}{cc}mathbf{8} & mathbf{2} \ mathbf{- 1} & -mathbf{3}end{array}right) )
find ( mathbf{6} boldsymbol{A}-mathbf{3} boldsymbol{B} )
12
291 If ( A ) and ( B ) are square matrices of order 3 such that ( |boldsymbol{A}|=-mathbf{1},|boldsymbol{B}|=mathbf{3}, ) then the
determinant of ( 3 A B ) is equal to
A . -9
B. -27
c. -81
D. 81
12
292 Find ( p, q, r ) and ( s, ) if ( left[begin{array}{cc}boldsymbol{p}+boldsymbol{4} & boldsymbol{2} boldsymbol{q}-boldsymbol{7} \ boldsymbol{s}-boldsymbol{3} & boldsymbol{r}+boldsymbol{2} boldsymbol{s}end{array}right]=left[begin{array}{cc}boldsymbol{6} & -boldsymbol{3} \ boldsymbol{2} & boldsymbol{1} boldsymbol{4}end{array}right] ) 12
293 If ( boldsymbol{A}=left(begin{array}{c}4-2 \ 5-9end{array}right) ) and ( boldsymbol{B}=left(begin{array}{cc}8 & 2 \ -1 & -3end{array}right) ) find ( 6 A )
3B.
12
294 Construct a ( 2 times 3 ) matrix ( A=left[a_{i j}right] )
whose elements are given by ( a_{i j}= )
( |2 i-3 j| )
12
295 If ( left[begin{array}{ccc}3 & 2 & -1 \ 4 & 9 & 2 \ 5 & 0 & -2end{array}right]left[begin{array}{l}x \ y \ zend{array}right]=left[begin{array}{l}0 \ 7 \ 2end{array}right], ) then
( (x, y, z)= )
В. (2,-1,-4)
c. (3,0,6)
( mathbf{D} cdot(2,-1,4) )
12
296 Find matrix ( boldsymbol{X}, ) if ( left[begin{array}{ccc}mathbf{3} & mathbf{5} & -mathbf{9} \ -mathbf{1} & mathbf{4} & -mathbf{7}end{array}right]+boldsymbol{X}= )
( left[begin{array}{lll}mathbf{6} & mathbf{2} & mathbf{3} \ mathbf{4} & mathbf{8} & mathbf{6}end{array}right] )
12
297 The number of ( boldsymbol{A} ) in ( boldsymbol{T}_{boldsymbol{p}} ) such that ( boldsymbol{A} ) is either symmetric or skew-symmetric or both, and det(A) divisible by ( p ) is:
A ( cdot(p-1)^{2} )
в. ( 2(p-1) )
c. ( (p-1)^{2}+1 )
D. ( 2 p-1 )
12
298 If ( A=left[begin{array}{cc}-3 & 5 \ 5 & 0 \ -7 & 4end{array}right] ) and ( B=left[begin{array}{ccc}3 & -5 & 7 \ -5 & 0 & -4end{array}right] )
then find ( boldsymbol{A}+boldsymbol{B}^{boldsymbol{T}} )
( A cdot 0 )
в. ( 2 B )
( c cdot 2 A^{T} )
D. ( 2 B^{T} )
12
299 Construct a ( 2 times 2 ) matrix, ( A=left[a_{i j}right] )
whose elements are given by:
(i) ( a_{i j}=frac{(i+j)^{2}}{2} )
(ii) ( a_{i j}=frac{i}{j} )
(iii) ( a_{i j}=frac{(i+2 j)^{2}}{2} )
12
300 ( operatorname{Let} A=left[begin{array}{ll}a & b \ c & dend{array}right], a, b, c, d neq 0, ) then
( boldsymbol{B}=boldsymbol{A} boldsymbol{A}^{prime}-boldsymbol{A}^{prime} boldsymbol{A} ) equals
A. ( (a d-b c) I )
в. ( (a c-b d) I )
c.
D. none of these
12
301 A matrix having ( m ) rows and ( n ) columns
with ( m neq n ) is said to be a
A. rectangular matrix
B. square matrix
c. identity matrix
D. scalar matrix
12
302 Solve the following system of equations by using Matrix inversion method.
( mathbf{2} boldsymbol{x}-boldsymbol{y}+mathbf{3} boldsymbol{z}=mathbf{9}, boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{6}, boldsymbol{x}- )
( boldsymbol{y}+boldsymbol{z}=mathbf{2} )
12
303 Assertion
The matrix ( left(begin{array}{cccc}.1 & 0 & 0 & 0 \ 0 & 2 & 0 & 0 \ 0 & 0 & 3 & 0end{array}right) ) is a
diagonal matrix
Reason
( A=left(a_{i j}right)_{m times m} ) is a square matrix such
that entry ( a_{i j}=0 forall i neq j, ) then ( A ) is
called diagonal matrix.
A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
C. (A) is true but (R)is false,
D. (A) is false but (R ) is true.
12
304 Choose the correct statement or
statements:

This question has multiple correct options
A. every scalar matrix is an identity matrix
B. every identity matrix is a scalar matrix
C. transpose of transpose of a matrix gives the matrix ¡tself
D. for every square matrix A there exists another matrix B such that ( A B=I=B A )

12
305 ( left[begin{array}{lll}boldsymbol{x} & mathbf{1} & -mathbf{1}end{array}right]left[begin{array}{lll}mathbf{0} & mathbf{1} & -mathbf{1} \ mathbf{2} & mathbf{1} & mathbf{3} \ mathbf{1} & mathbf{1} & mathbf{1}end{array}right]left[begin{array}{l}boldsymbol{x} \ -mathbf{1} \ mathbf{1}end{array}right]=mathbf{0} )
then find ( x )
12
306 If ( A ) is an invertible matrix of order ( 3 times 3 )
such that ( |boldsymbol{A}|=mathbf{5}, ) find the value of
( left|boldsymbol{A}^{-1}right| )
12
307 If ( boldsymbol{A}=left[boldsymbol{a}_{i j}right]_{2 times 2} ) and ( boldsymbol{a}_{i j}=boldsymbol{i}+boldsymbol{j}, ) then ( boldsymbol{A}= )
( A cdotleft(begin{array}{l}12 \ 3end{array}right) )
в. ( left(begin{array}{c}23 \ 3end{array}right) )
( c cdotleft(begin{array}{cc}2 & 3 \ 4 & 5end{array}right) )
D. ( left(_{6}^{4} begin{array}{r}5 \ end{array}right) )
12
308 ( operatorname{Let} C_{k}=^{n} C_{k} ) for ( 0 leq k leq n ) and
( boldsymbol{A}_{boldsymbol{k}}=left[begin{array}{cc}boldsymbol{C}_{boldsymbol{k}-1}^{2} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{C}_{boldsymbol{k}}^{2}end{array}right] ) for ( boldsymbol{k} geq 1, ) and ( boldsymbol{A}_{1}+ )
( boldsymbol{A}_{2}+ldots+boldsymbol{A}_{n}=left[begin{array}{cc}boldsymbol{k}_{1} & mathbf{0} \ mathbf{0} & boldsymbol{k}_{2}end{array}right], ) then
This question has multiple correct options
A ( . k_{1}=k_{2} )
B . ( k_{1}+k_{2}=^{2 n} C_{2 n}+1 )
c. ( k_{1}=^{2 n} C_{n}-1 )
D. ( k_{2}=^{2 n} C_{n+1} )
12
309 Let ( A, B, C, D ) be (not necessarily
square) real matrices such that ( boldsymbol{A}^{boldsymbol{T}}= )
( boldsymbol{B C D} ; boldsymbol{B}^{boldsymbol{T}}=boldsymbol{C D A} ; boldsymbol{C}^{boldsymbol{T}}=boldsymbol{D A B} ) and
( D^{T}=A B C ) for the matrix ( S=A B C D )
then which of the following is/are true This question has multiple correct options
A ( cdot S^{3}=S )
B. ( S^{2}=S^{4} )
c. ( S=S^{2} )
D. none of these
12
310 ( mathbf{f}left[begin{array}{cc}boldsymbol{x}-boldsymbol{y} & boldsymbol{z} \ mathbf{2} boldsymbol{x}-boldsymbol{y} & boldsymbol{w}end{array}right]=left[begin{array}{cc}-mathbf{1} & mathbf{4} \ mathbf{0} & mathbf{5}end{array}right], ) find the
value of ( boldsymbol{x}+boldsymbol{y} )
12
311 Multiplication of 3 with the matrix ( left[begin{array}{lll}mathbf{4} & mathbf{1} & mathbf{1}end{array}right] ) gives
( mathbf{A} cdotleft[begin{array}{lll}12 & 3 & 3end{array}right] )
B cdot ( left[begin{array}{lll}12 & 1 & 1end{array}right] )
( mathbf{c} cdotleft[begin{array}{lll}4 & 3 & 1end{array}right] )
D. ( left[begin{array}{lll}4 & 1 & 3end{array}right] )
12
312 Let ( M ) and ( N ) be two ( 3 times 3 ) non-singular
skew-symmetric matrices such that
( M N=N M . ) If ( P^{T} ) denotes the
transpose of ( P, ) then
( M^{2} N^{2}left(M^{T} Nright)^{-1}left(M N^{-1}right)^{T} ) is equal to
A ( . M^{2} )
B. ( -N^{2} )
c. ( -M^{2} )
D. ( M N )
12
313 Let ( a ) denote the element of the ( i^{t h} ) row
and ( j^{t h} ) column in a ( 3 times 3 ) matrix and let
( boldsymbol{a}_{i j}=-boldsymbol{a}_{j i} ) for every i and j then this
matrix is an –
A. Orthogonal matrix
B. singular matrix
c. matrix whose principal diagonal elements are all zero
D. skew-symmetric matrix
12
314 Let ( A ) be a real matrix such that ( A^{67}= )
( A^{-1}, ) then
( mathbf{A} cdot|A|=pm 1 )
B . ( |A|=1 )
C. ( A=I, I ) being unit matrix
D. ( A ) is diagonal matrix
12
315 Find the values of ( x ), if
( left|begin{array}{ll}2 & 4 \ 5 & 1end{array}right|=left|begin{array}{cc}2 x & 4 \ 6 & xend{array}right| )
12
316 If ( boldsymbol{P}=left[begin{array}{ccc}boldsymbol{a} & boldsymbol{0} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{b} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{0} & boldsymbol{c}end{array}right] ) then, ( operatorname{det}left(boldsymbol{P}^{-1}right) )
A ( . a b c )
B. ( a^{2} b^{2} c^{2} )
c. ( frac{1}{a b} )
D. ( frac{1}{a^{2} b^{2} c^{2}} )
12
317 ( operatorname{Let} boldsymbol{A}=left[begin{array}{ll}mathbf{3} & mathbf{7} \ mathbf{2} & mathbf{5}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{ll}mathbf{6} & mathbf{8} \ mathbf{7} & mathbf{9}end{array}right] . ) Verify
( operatorname{that}(A B)^{-1}=B^{-1} A^{-1} )
12
318 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{3} & mathbf{- 2} \ mathbf{5} & mathbf{4}end{array}right], boldsymbol{B}=left[begin{array}{cc}mathbf{1} & mathbf{4} \ mathbf{6} & -mathbf{7}end{array}right], ) then
find the matrix ( A-4 B+7 I ), where lis
the unit matrix of order 2
( =left(begin{array}{rr}3 & -2 \ 5 & 4end{array}right], mathbf{B}=left[begin{array}{rr}1 & 4 \ 6 & -7end{array}right] )
12
319 If order of a matrix is ( 3 times 3, ) then it is a
A. square matrix
B. rectangular matrix
c. unit matrix
D. None of these
12
320 ( mathbf{A}=left[begin{array}{cc}cos alpha & sin alpha \ -sin alpha & cos alphaend{array}right] ) then ( mathbf{A} . mathbf{A}^{mathbf{T}} )
A. Null matrix
в. А
( c cdot I )
D. A
12
321 If ( boldsymbol{A}=left[begin{array}{cc}-mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{4}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{cc}mathbf{3} & -mathbf{2} \ mathbf{1} & mathbf{5}end{array}right] )
( mathbf{2} boldsymbol{A}+boldsymbol{B}+boldsymbol{X}=mathbf{0}, ) then the matrix ( boldsymbol{X} ) is
A. ( left[begin{array}{ll}1 & 2 \ 7 & 13end{array}right] )
B. ( left[begin{array}{cc}-1 & 2 \ 7 & -13end{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}-1 & -2 \ 7 & 13end{array}right] )
D. ( left[begin{array}{cc}-1 & -2 \ -7 & -13end{array}right] )
12
322 ( mathrm{ff} mathbf{A}=left[begin{array}{lll}1 & 2 & x \ 0 & 1 & 0 \ 0 & 0 & 1end{array}right] ) and ( B= )
( left[begin{array}{ccc}1 & -2 & y \ 0 & 1 & 0 \ 0 & 0 & 1end{array}right] quad ) and
( A B=I_{3}, quad ) then ( quad x+ )
( boldsymbol{y} quad boldsymbol{i s} quad boldsymbol{e q u a l} quad boldsymbol{t o} )
12
323 If the matrix ( left|begin{array}{ccc}mathbf{1} & mathbf{3} & boldsymbol{lambda}+mathbf{2} \ mathbf{2} & mathbf{4} & mathbf{8} \ mathbf{3} & mathbf{5} & mathbf{1 0}end{array}right| ) is singular
( operatorname{then} lambda= )
( A cdot-2 )
B. 4
( c cdot 2 )
D. – –
12
324 Let ( A ) be a square matrix all of whose
entries are integers. Then, which one of
the following is true?
( mathbf{A} cdot ) If ( operatorname{det} A=pm 1, ) then ( A^{-1} ) exists and all its entries are
integers
B. If ( operatorname{det} A=pm 1, ) then ( A^{-1} ) need not exist
C. If ( operatorname{det} A=pm 1, ) then ( A^{-1} ) exists but all its entries are
not necessarily integers
D. If det ( A neqpm 1 ), then ( A^{-1} ) exists and all its entries are non-integers
12
325 Is it possible to define the matrix AB and BA when :
A has 4 rows and ( mathrm{B} ) has 4 columns
12
326 Using elementary operations, find the inverse of the following matrix:
( left(begin{array}{ccc}-1 & 1 & 2 \ 1 & 2 & 3 \ 3 & 1 & 1end{array}right) )
12
327 In the set of all ( 3 times 3 ) real matrices a
relation is defined as follows. A matrix
( A ) is related to a matrix ( B ), if and only
there is a non-singular ( 3 times 3 ) matrix ( P )
such that ( B=P^{-1} A P . ) This relation is
A. reflexive,symmetric but not transitive
B. reflexive, transitive but not symmetric
c. symmetric, transitive but not reflexive
D. an equivalence relation
12
328 Matrices ( A ) and ( B ) satisfy ( A B=B^{-1} ) where ( boldsymbol{B}=left[begin{array}{cc}2 & -1 \ 2 & 0end{array}right], ) then
find without finding ( A^{-1}, ) the matrix ( X )
satisfying ( boldsymbol{A}^{-1} boldsymbol{X} boldsymbol{A}=? )
A. ( B )
в. ( B^{2} )
c. ( A )
D. None of these
12
329 If ( boldsymbol{m}left[begin{array}{ll}-mathbf{3} & mathbf{4}end{array}right]+boldsymbol{n}left[begin{array}{ll}mathbf{4} & -mathbf{3}end{array}right]=left[begin{array}{ll}mathbf{1 0} & -mathbf{1 1}end{array}right] )
then ( 3 m+7 n= )
( A cdot 3 )
B. 5
c. 10
( D )
12
330 If ( boldsymbol{A}=left(left[begin{array}{llll}1 & 2 & 3 & 4end{array}right] text { and } A B=left[begin{array}{lll}3 & 4 & -1end{array}right]right. )
then the order of
matrix B is
( A cdot 2 times 3 )
B. 3×3
( c cdot 4 times 3 )
D. ( 1 times 3 )
12
331 Assertion ( operatorname{Let} A=left[begin{array}{ll}a & b \ b & aend{array}right] ) and ( B=left[begin{array}{ll}p & q \ r & send{array}right] )
If ( b=0, ) then ( A B=B A )
Reason

If ( b neq 0, ) then ( A B=B A )
( Leftrightarrow boldsymbol{p}=boldsymbol{s} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{q}=boldsymbol{r} )
A. Both Assertion and Reason are correct and Reason is
the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct

12
332 By using the elementary
transformation, find the inverse of the ( operatorname{matrix} boldsymbol{A}=left[begin{array}{cc}mathbf{1} & -mathbf{2} \ mathbf{2} & mathbf{1}end{array}right] )
12
333 If ( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{4} & mathbf{7} \ mathbf{2} & mathbf{6} & mathbf{5} \ mathbf{3} & mathbf{- 1} & mathbf{2}end{array}right] ) and ( mathbf{B}=operatorname{diag}(12 )
5), then
trace of matrix ( boldsymbol{A B}^{mathbf{2}} ) is
A . 74
B. 75
c. 529
D. 23
12
334 By using elementary transformation
find the inverse of the matrix ( boldsymbol{A}= ) ( left[begin{array}{ll}1 & 2 \ 2 & 1end{array}right] )
12
335 If ( [123] mathrm{B}=[34], ) the order of the matrix B is
( mathbf{A} cdot 3 times 1 )
B. ( 1 times 3 )
c. ( 2 times 3 )
D. ( 3 times 2 )
12
336 The management committee of a residential colony decided to award some of its members (say ( x ) ) for
honesty, some (say ( y ) ) for helping others
and some others (say ( z ) ) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12
Three times the sum of awardees for
cooperation and supervision added to two times the number of awardees for
honesty is ( 33 . ) If the sum of the number of awardees for honesty and supervision
is twice the number of awardees for
helping others using matrix category. Apart from these values namely, honesty, cooperation and supervision,
suggest one more value which the
management of the colony must include for awards.
12
337 Find the value of ( y, ) if ( left[begin{array}{cc}x-y & 2 \ x & 5end{array}right]= )
( left[begin{array}{ll}2 & 2 \ 3 & 5end{array}right] )
12
338 Solve the following matrix equation for
( x )
( [boldsymbol{x}-mathbf{5}-mathbf{1}]left[begin{array}{lll}mathbf{1} & mathbf{0} & mathbf{2} \ mathbf{0} & mathbf{2} & mathbf{1} \ mathbf{2} & mathbf{0} & mathbf{3}end{array}right]left[begin{array}{l}boldsymbol{x} \ mathbf{4} \ mathbf{1}end{array}right]=mathbf{0} )
12
339 Solve for ( x ) and ( y, ) if ( left(begin{array}{l}x^{2} \ y^{2}end{array}right)+3left(begin{array}{c}2 x \ -yend{array}right)= )
( left(begin{array}{c}-mathbf{9} \ mathbf{4}end{array}right) )
12
340 If the matrix ( A B ) is a zero matrix, then
which one of the following is correct?
A. ( A ) must be equal to zero matrix or ( B ) must be equal to zero matrix
B. ( A ) must be equal to zero matrix and ( B ) must be equal to zero matrix
c. It is not necessary that either ( A ) is zero matrix or ( B ) is zero matrix
D. None of the above
12
341 ( boldsymbol{A}=left[begin{array}{ll}mathbf{0} & mathbf{1} \ mathbf{0} & mathbf{0}end{array}right], ) show that ( (boldsymbol{a} boldsymbol{I}+boldsymbol{b} boldsymbol{A})^{n}= )
( a^{n} I+n a^{n-1} b A, ) where ( I ) is the identity
matrix of order 2 and ( n in N )
12
342 Using elementary transformations, find the inverse of matrix, ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{3} \ mathbf{2} & mathbf{7}end{array}right] ) 12
343 If ( A ) is an invertible matrix of order 2
then ( operatorname{det}left(A^{-1}right) ) is equal to
( mathbf{A} cdot operatorname{det}(A) )
B. ( frac{1}{operatorname{det}(A)} )
( c .1 )
D. 0
12
344 The order of the matrix ( left[begin{array}{c}-1 \ 3 \ 4end{array}right] ) is :
( mathbf{A} cdot 1 times 3 )
B. ( 3 times 1 )
c. ( 1 times 1 )
D. ( 3 times 3 )
12
345 If for a matrix ( boldsymbol{A}, boldsymbol{A}^{2}+boldsymbol{I}=boldsymbol{O} ) where ( boldsymbol{I} ) is
the identity matrix, then ( A ) equals
( mathbf{A} cdotleft[begin{array}{ll}1 & 0 \ 0 & 1end{array}right] )
В. ( left[begin{array}{cc}i & 0 \ i & -iend{array}right] )
c. ( left[begin{array}{cc}1 & 2 \ -1 & 1end{array}right] )
D. ( left[begin{array}{cc}-1 & 0 \ 0 & -1end{array}right] )
12
346 If ( A ) is non-singular and ( (A+I)(A- )
( mathbf{3} boldsymbol{I})=mathbf{0} ) then ( mathbf{3} boldsymbol{A}^{-1}-boldsymbol{A}+mathbf{2} boldsymbol{I} )
A . ( I )
B.
c. ( 2 I )
D. ( 6 I )
12
347 Prove that ( left|begin{array}{ccc}boldsymbol{y} boldsymbol{z}-boldsymbol{x}^{2} & boldsymbol{z} boldsymbol{x}-boldsymbol{y}^{2} & boldsymbol{x} boldsymbol{y}-boldsymbol{z}^{2} \ boldsymbol{z} boldsymbol{x}-boldsymbol{y}^{2} & boldsymbol{x} boldsymbol{y}-boldsymbol{z}^{2} & boldsymbol{y} boldsymbol{z}-boldsymbol{x}^{2} \ boldsymbol{x} boldsymbol{y}-boldsymbol{z}^{2} & boldsymbol{y} boldsymbol{z}-boldsymbol{x}^{2} & boldsymbol{z} boldsymbol{x}-boldsymbol{y}^{2}end{array}right| ) is
divisible by ( (x+y+z), ) and hence find
the quotient
OR
Using elementary transformations, find the inverse of the matrix ( boldsymbol{A}= ) ( left(begin{array}{lll}8 & 4 & 3 \ 2 & 1 & 1 \ 1 & 2 & 2end{array}right) ) and use it to solve the
following system of linear equations:
( 8 x+4 y+3 z=19 )
( 2 x+y+z=5 )
( x+2 y+2 z=7 )
12
348 If ( boldsymbol{A}=[mathbf{1}], ) then the order of the matrix
is
( mathbf{A} cdot 1 times 1 )
в. ( 2 times 1 )
c. ( 1 times 2 )
D. None of these
12
349 Convert ( [1 quad-12 )
3] into an identity
matrix by suitable row transformations.
12
350 If ( A ) and ( B ) are skew symmetric
matrices of order ( n ) then ( A+B ) is
A. skew symmetric
B. a diagonal matrix
c. a null matrix
D. symmetric matrix
12
351 If ( x ) and ( Y ) are the matrices of order ( 2 x )
2 each and ( 2 X-3 Y=left|begin{array}{cc}-7 & 0 \ 7 & -13end{array}right| ) and
( mathbf{3} boldsymbol{X}+mathbf{2} boldsymbol{Y}=left|begin{array}{cc}mathbf{9} & mathbf{1 3} \ mathbf{4} & mathbf{1 3}end{array}right|, ) then what is ( boldsymbol{Y} )
equal to?
A. ( left|begin{array}{cc}1 & 3 \ -2 & 1end{array}right| )
B. ( left|begin{array}{ll}1 & 3 \ 2 & 1end{array}right| )
begin{tabular}{l|ll|l}
& 3 & 2 \
-1 & 5
end{tabular}
D. ( left|begin{array}{cc}3 & 2 \ 1 & -5end{array}right| )
12
352 If ( A=left|begin{array}{l}1 \ 3end{array}right| B=left|begin{array}{c}-1 \ 4end{array}right| ) then ( 2 A+B= )
( A cdot mid 10 )
9
B. ( mid 10 ) 1
( c cdotleft|begin{array}{l}1 \ 10end{array}right| )
D. ( mid 1 ) 9
12
353 For square matrix ( boldsymbol{A}, boldsymbol{A} boldsymbol{A}^{boldsymbol{T}} ) is –
A. unit matrix
B. symmetric matrix
c. skew symmetric matrix
D. diagonal matrix
12
354 If ( mathbf{A} ) is ( 3 x 4 ) matrix ( B^{T} ) is a matrix such
that ( A^{T} B ) and ( B A^{T} ) are both defined
then ( B ) is of the type
( A cdot 3 times 4 )
B. ( 3 times 3 )
c. ( 4 times 3 )
D. ( 4 times 4 )
12
355 type of entertainment device sold at
three of their branch stores so that they
can monitor their purchases of
supplies. The sales in two weeks are
shown in the following spreadsheets.
Find the sum of the items sold out in
two weeks using matrix addition.
12
356 Construct a ( 4 times 3 ) matrix ( A=left[a_{i j}right] ) whose element ( a_{i j} ) is ( a_{i j}=2 i+frac{i}{j} ) 12
357 ( mathbf{f} boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{0} & mathbf{0} \ mathbf{0} & mathbf{1} & mathbf{1} \ mathbf{0} & -mathbf{2} & mathbf{4}end{array}right], boldsymbol{I}=left[begin{array}{ccc}mathbf{1} & mathbf{0} & mathbf{0} \ mathbf{0} & mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{0} & mathbf{1}end{array}right] )
and
( boldsymbol{A}^{-1}=frac{1}{boldsymbol{6}}left(boldsymbol{A}^{2}+boldsymbol{alpha} boldsymbol{A}+boldsymbol{beta} boldsymbol{I}right), ) find ( boldsymbol{beta} / mathbf{1 1} )
12
358 Find ( boldsymbol{A}+boldsymbol{B} )
( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{3} & mathbf{- 1} \ mathbf{2} & mathbf{- 5} & mathbf{4} \ mathbf{3} & mathbf{2} & mathbf{- 6}end{array}right] ) and ( boldsymbol{B}= )
( left[begin{array}{ccc}mathbf{2} & mathbf{4} & mathbf{0} \ mathbf{3} & mathbf{- 4} & mathbf{5} \ mathbf{2} & mathbf{3} & mathbf{- 5}end{array}right] )
12
359 ( boldsymbol{A}=left[begin{array}{cc}boldsymbol{t} & boldsymbol{t}+mathbf{1} \ boldsymbol{t}-mathbf{1} & boldsymbol{t}end{array}right] ) is a matrix such
that ( A A^{T}=I_{2} ) then trace of the matrix
is
A . 2
B. 0
( c cdot 4 )
( D )
12
360 Prove the following ( left[begin{array}{llll}1 & 3 & 2 & 0 \ 4 & 1 & 5 & 9 \ 3 & 2 & 1 & 3end{array}right] pm )
( left[begin{array}{cccc}1 & 0 & 5 & 8 \ 2 & 3 & 5 & 8 \ 1 & -5 & 2 & 3end{array}right] )
( =left[begin{array}{cccc}2 & 3 & 7 & 8 \ 6 & 4 & 9 & 14 \ 4 & -3 & 3 & 6end{array}right] ) for plus
( =left[begin{array}{cccc}0 & 3 & -3 & -8 \ 4 & 2 & 1 & 4 \ 0 & 7 & -1 & 0end{array}right] ) for minus
12
361 By using elementary transformation find the inverse of ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{2}-mathbf{1}end{array}right] ) 12
362 For a matrix ( boldsymbol{A}left(begin{array}{ccc}mathbf{1} & mathbf{0} & mathbf{0} \ mathbf{2} & mathbf{1} & mathbf{0} \ mathbf{3} & mathbf{2} & mathbf{1}end{array}right), ) if ( boldsymbol{U}_{1}, boldsymbol{U}_{2} )
and ( U_{3} ) are ( 3 times 1 ) column matrices
satisfying ( boldsymbol{A} boldsymbol{U}_{mathbf{1}}= )
( left(begin{array}{l}1 \ 0 \ 0end{array}right), A U_{2}left(begin{array}{l}2 \ 3 \ 0end{array}right), A U_{3}=left(begin{array}{l}2 \ 3 \ 1end{array}right) ) and ( U ) is
( 3 times 3 ) matrix whose columns are ( U_{1}, U_{2} )
and ( U_{3} )
Then sum of the elements of ( U^{-1} ) is
A . 6
B. ( 0(z e r o) )
( c . )
D. ( 2 / 3 )
12
363 If ( A ) is non-singular and ( (A-2 I)(A- ) ( 4 I)=0 ) then ( frac{1}{6} mathbf{A}+frac{4}{3} mathbf{A}^{-1}= )
( A cdot I )
B.
c. ( 2 I )
D. ( 6 I )
12
364 If ( A ) is a diagonal matrix of order ( 3 times 3 )
is a commutative with every square
matrix of order ( 3 times 3 ) under
multiplication and ( operatorname{tr}(A)=12, ) then the value of ( |boldsymbol{A}| ) is
12
365 ( A=left[begin{array}{ll}1 & 2 \ 3 & 4end{array}right], B=left[begin{array}{ll}-1 & -1 \ -1 & -1end{array}right], C=left[begin{array}{ll}x & y \ z & rend{array}right] )
If ( boldsymbol{A}+mathbf{3} boldsymbol{B}=boldsymbol{C}, ) then ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}+boldsymbol{r} ) is
( A )
B. – 2
( c .-1 )
( D )
12
366 If ( boldsymbol{P}=left[begin{array}{cc}frac{sqrt{mathbf{3}}}{mathbf{2}} & frac{mathbf{1}}{2} \ -frac{mathbf{1}}{mathbf{2}} & frac{sqrt{mathbf{3}}}{mathbf{2}}end{array}right], boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{1} \ mathbf{0} & mathbf{1}end{array}right] ) and
( Q=P A P^{T} ) then ( Pleft(Q^{2005}right) P^{T} ) equal to
begin{tabular}{l}
A. ( left[begin{array}{cc}1 & 2005 \
0 & 1end{array}right] ) \
hline
end{tabular}
B. ( left[begin{array}{cc}sqrt{3} / 2 & 2005 \ 1 & 0end{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}1 & 2005 \ sqrt{3} / 2 & 1end{array}right] )
D. ( left[begin{array}{ll}1 & sqrt{3} / 2 \ 0 & 2005end{array}right] )
12
367 If ( A ) and ( B ) are two non-zero square
matrices of the same order such that
the product ( A B=0, ) then
A. both A and B must be singular
B. exactly one of them must be singular
c. atleast one of them must be non-singular
D. none of these
12
368 ( left[begin{array}{lll}0 & 0 & 0end{array}right] ) is an example of
A. Scalar matrix
B. Diagonal matrix
c. Identity matrix
D. Null matrix
12
369 If ( mathbf{3} boldsymbol{A}-mathbf{2 B}=left(begin{array}{cc}mathbf{1} & mathbf{- 2} \ mathbf{3} & mathbf{0}end{array}right) ) and ( mathbf{2} boldsymbol{A}- )
( mathbf{3} boldsymbol{B}=left(begin{array}{cc}mathbf{- 3} & mathbf{3} \ mathbf{1} & mathbf{- 1}end{array}right) ) then find ( boldsymbol{B} )
12
370 ( operatorname{Let} P=left(begin{array}{cc}cos frac{pi}{4} & -sin frac{pi}{4} \ sin frac{pi}{4} & cos frac{pi}{4}end{array}right) ) and ( x= )
( left(begin{array}{c}frac{1}{sqrt{2}} \ frac{1}{sqrt{2}}end{array}right) cdot operatorname{Then} P^{3} X ) is equal to
A ( cdotleft(begin{array}{l}0 \ 1end{array}right) )
B. ( left(frac{-1}{sqrt{2}}right) )
( left(begin{array}{c}-1 \ 0end{array}right) )
D ( left(begin{array}{l}-frac{1}{sqrt{2}} \ -frac{1}{sqrt{2}}end{array}right) )
12
371 ( operatorname{Let} boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{0} \ mathbf{1} & mathbf{1}end{array}right], ) and ( boldsymbol{I}=left[begin{array}{ll}mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{1}end{array}right] ) then
prove that ( boldsymbol{A}^{n}=boldsymbol{n} boldsymbol{A}-(boldsymbol{n}-mathbf{1}) boldsymbol{I}, boldsymbol{n} geq 1 )
12
372 Find the inverse of the matrix ( boldsymbol{A}= ) ( left[begin{array}{ll}1 & 2 \ 1 & 3end{array}right] ) using elementry
transformations.
12
373 ( operatorname{Given} boldsymbol{A}=left[begin{array}{cc}mathbf{3} & mathbf{6} \ -mathbf{2} & -mathbf{8}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{ll}mathbf{2} & mathbf{1 6}end{array}right] )
find the matrix ( boldsymbol{X} ) such that ( boldsymbol{X} boldsymbol{A}=boldsymbol{B} )
A ( cdotleft[-frac{4}{3}-3right] )
в. ( left[frac{4}{3} 3right] )
( ^{mathbf{c}} cdotleft[begin{array}{ll}frac{4}{3} & -3end{array}right] )
D. ( left[-frac{4}{3} quad 3right] )
12
374 If ( boldsymbol{A} ) is symmetric matrix and ( boldsymbol{n} in boldsymbol{N} )
write whether ( A^{n} ) is symmetric or skewsymmetric or neither of these two
12
375 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{1} \ mathbf{2} & mathbf{2}end{array}right], boldsymbol{B}=left[begin{array}{ll}mathbf{1} & mathbf{1} \ mathbf{3} & mathbf{4}end{array}right] ) then find
( boldsymbol{A B} )
12
376 Matrices ( A ) and ( B ) satisfy ( A B=B^{-1} ) where ( boldsymbol{B}=left[begin{array}{cc}2 & -1 \ 2 & 0end{array}right], ) then
find without finding ( B^{-1}, ) the value of ( K )
for which ( boldsymbol{K} boldsymbol{A}-mathbf{2} boldsymbol{B}^{-mathbf{1}}+boldsymbol{I}=boldsymbol{O} )
12
377 ( left[begin{array}{ccc}10 & 20 & 30 \ 20 & 45 & 80 \ 30 & 80 & 171end{array}right]= )
( left[begin{array}{lll}1 & 0 & 0 \ 2 & 1 & 0 \ 3 & 4 & 1end{array}right]left[begin{array}{lll}x & 0 & 0 \ 0 & 5 & 0 \ 0 & 0 & 1end{array}right]left[begin{array}{lll}1 & 2 & 3 \ 0 & 1 & 4 \ 0 & 0 & 1end{array}right] ) then
( boldsymbol{x}= )
( A cdot 10 )
B. 20
( c cdot 30 )
D. 40
12
378 If ( A=left[begin{array}{ll}alpha & 0 \ 1 & 1end{array}right], B=left[begin{array}{ll}1 & 0 \ 5 & 1end{array}right] ) whenever
( boldsymbol{A}^{2}=boldsymbol{B} )
then values of ( boldsymbol{alpha} ) is
A .
B. –
( c cdot 4 )
D. no real value of ( alpha )
12
379 Find the value of ( x+y ) from the
following equation:
( mathbf{2}left[begin{array}{cc}boldsymbol{x} & mathbf{5} \ mathbf{7} & boldsymbol{y}-mathbf{3}end{array}right]+left[begin{array}{cc}mathbf{3} & mathbf{- 4} \ mathbf{1} & mathbf{2}end{array}right]=left[begin{array}{cc}mathbf{7} & mathbf{6} \ mathbf{1 5} & mathbf{1 4}end{array}right] )
12
380 Let ( A ) be the set of all ( 3 times 3 ) skew
symmetric matrices whose entries are
either ( -1,0, ) or ( 1 . ) If there are exactly
three 0 s, three 1 s, and three (-1) s, then the number of such matrices is
12
381 Assertion
If ( A ) and ( B ) are two ( 3 times 3 ) matrices such
that ( A B=0, ) then ( A=0 ) or ( B=0 )
Reason
If ( A, B ) and ( X ) are three ( 3 times 3 ) matrices
such that ( boldsymbol{A} boldsymbol{X}=boldsymbol{B},|boldsymbol{A}| neq mathbf{0}, ) then ( boldsymbol{X}= )
( A^{-1} B )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
382 If the matrices has 13 elements , then
the possible dimension (order) it can have are
A . ( 1 times 13 ) or ( 13 times 1 )
B. ( 1 times 26 ) or ( 26 times 1 )
c. ( 2 times 13 ) or ( 13 times 2 )
D. None of these
12
383 ( mathbf{f} boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{1}end{array}right] ; boldsymbol{B}=left[begin{array}{ll}boldsymbol{a} & boldsymbol{c} \ boldsymbol{b} & boldsymbol{d}end{array}right], ) then show
that ( (A . B)^{-1}=B^{-1} . A^{-1} )
12
384 11.
13 -1 -2]
Let P= 2 0 a
[3 -5 o
matrix such that PQ = kl, where k ER,k
te a ER. Suppose Q=[q;] is a
u
wherek eRk+ 0 and I is the
k2
and det(Q) =
identity matrix of order 3. If 423 = -3 and
then
(JEE Adv. 2016)
(a) a= 0, k=8
(b) 4a-k+8=0
© det (Padj (Q))=29 (d) det (Q adj (P))=2
12
385 Find the inverse of ( boldsymbol{A}= )
( left[begin{array}{ccc}cos alpha & -sin alpha & 0 \ sin alpha & cos alpha & 0 \ 0 & 0 & 1end{array}right] ) using elementary
transformation.
12
386 For the matrix ( boldsymbol{A}=left[begin{array}{ccc}mathbf{4} & -mathbf{4} & mathbf{5} \ -mathbf{2} & mathbf{3} & -mathbf{3} \ mathbf{3} & -mathbf{3} & mathbf{4}end{array}right] ) 12
387 State true or false:
The determinant of a skew-symmetric matrix is a perfect square if it’s
elements are integers.
A. True
B. False
12
388 Find the output order for the following
matrix multiplication ( boldsymbol{A}_{mathbf{4} times mathbf{2}} times boldsymbol{B}_{mathbf{2} times mathbf{4}} ) ?
( A cdot 2 times 4 )
B. ( 4 times 4 )
( c cdot 4 times 2 )
D. Multiplication not possible
12
389 Let ( A ) be the ( 2 times 2 ) matrices given by
( boldsymbol{A}=left[boldsymbol{a}_{i j}right] ) where ( boldsymbol{a}_{i j}={mathbf{0}, mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}} )
such that ( boldsymbol{a}_{11}+boldsymbol{a}_{12}+boldsymbol{a}_{21}+boldsymbol{a}_{22}=boldsymbol{4} )
Find the number of matrices ( A ) such
that the trace of ( A ) is equal to 4
( A cdot 3 )
B. 4
( c cdot 5 )
D. 6
12
390 What is the result when you add the ( operatorname{matrix}left[begin{array}{ll}mathbf{4} & mathbf{5}end{array}right] ) to the matrix ( left[begin{array}{ll}mathbf{7} & -mathbf{3}end{array}right] )
and multiply the result by ( 2 ? )
( mathbf{A} cdotleft[begin{array}{ll}2 & 6end{array}right] )
в. ( left[begin{array}{ll}11 & 2end{array}right] )
c. ( left[begin{array}{ll}22 & 4end{array}right] )
D. ( [28 quad-15] )
E . ( left[begin{array}{ll}54 & -30end{array}right] )
12
391 The value of ( x ) satisfying the equation 2 ( left|begin{array}{cc}mathbf{3} & mathbf{1} \ mathbf{1} & mathbf{2}end{array}right|+left|begin{array}{cc}boldsymbol{x}^{mathbf{2}} & mathbf{9} \ -mathbf{1} & mathbf{0}end{array}right|=left|begin{array}{cc}mathbf{5} boldsymbol{x} & mathbf{6} \ mathbf{0} & mathbf{1}end{array}right|+left|begin{array}{cc}mathbf{0} & mathbf{5} \ mathbf{1} & mathbf{3}end{array}right| ) are
( A cdot 1,2 )
B. 2,3
( c .pm 2 )
( mathrm{D} cdot pm 3 )
12
392 ( mathbf{f} boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{0} & mathbf{0} \ mathbf{0} & mathbf{1} & mathbf{1} \ mathbf{0} & -mathbf{2} & mathbf{4}end{array}right], mathbf{6} boldsymbol{A}^{-mathbf{1}}=boldsymbol{A}^{mathbf{2}}+ )
( c A+d I, ) then ( (c, d) ) is equal to
A ( cdot(-6,11) )
в. (-11,6)
c. (11,6)
( D )
12
393 ( left[begin{array}{cc}mathbf{1} & mathbf{1} \ mathbf{0} & mathbf{1}end{array}right]left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{0} & mathbf{1}end{array}right]left[begin{array}{ll}mathbf{1} & mathbf{3} \ mathbf{0} & mathbf{1}end{array}right] cdotleft[begin{array}{cc}mathbf{1} & boldsymbol{n}-mathbf{1} \ mathbf{0} & mathbf{1}end{array}right]= )
( left[begin{array}{cc}mathbf{1} & mathbf{7 8} \ mathbf{0} & mathbf{1}end{array}right] )
If ( boldsymbol{A}=left[begin{array}{cc}mathbf{1} & boldsymbol{n} \ mathbf{0} & mathbf{1}end{array}right] operatorname{then} boldsymbol{A}^{-mathbf{1}}=? )
A ( cdotleft[begin{array}{cc}1 & 12 \ 0 & 1end{array}right] )
B ( cdotleft[begin{array}{cc}1 & -13 \ 0 & 1end{array}right] )
c. ( left[begin{array}{cc}1 & -12 \ 0 & 1end{array}right] )
D. ( left[begin{array}{cc}1 & 0 \ -13 & 1end{array}right] )
12
394 Let ( mathbf{A} ) be a square matrix. Consider
( left.left.text { 1) }left.mathbf{A}+mathbf{A}^{mathbf{T}} text { 2 }right) mathbf{A} mathbf{A}^{mathbf{T}} mathbf{3}right) mathbf{A}^{mathbf{T}} mathbf{A} mathbf{4}right) mathbf{A}^{mathbf{T}}+mathbf{A} )
5) ( left.mathbf{A}-mathbf{A}^{mathbf{T}} mathbf{6}right) mathbf{A}^{mathbf{T}}-mathbf{A}, ) Then
A. all are symmetric matrices
B. (2),(4), (6) are symmetric matrices
c. (1),(2),(3),(4) are symmetric matrices &
(5),(6) are skew symmetric matrices
D. 5,6 are symmetric
12
395 Find ( : A^{2} ) if
( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{3} & mathbf{2} \ -mathbf{5} & mathbf{1} & mathbf{0} \ mathbf{1} & mathbf{2} & mathbf{5}end{array}right] )
12
396 If ( X, Y ) are two matrices given by the equations ( boldsymbol{X}+boldsymbol{Y}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{4}end{array}right], boldsymbol{X}-boldsymbol{Y}= )
( left[begin{array}{cc}mathbf{3} & mathbf{2} \ -mathbf{1} & mathbf{0}end{array}right] )
( operatorname{then} boldsymbol{X} boldsymbol{Y}=left[begin{array}{ll}boldsymbol{a} & boldsymbol{b} \ boldsymbol{c} & boldsymbol{d}end{array}right], ) find ( boldsymbol{a}-boldsymbol{b}+boldsymbol{c}+boldsymbol{d} )
12
397 IF ( A, B, C ) are non-singular ( n times n )
matrices, then ( (A B C)^{-1}= )
( mathbf{A} cdot A^{-1} C^{-1} B^{-1} )
B. ( C^{-1} B^{-1} A^{-1} )
c. ( C^{-1} A^{-1} B^{-1} )
D. ( B^{-1} C^{-1} A^{-1} )
12
398 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{5} boldsymbol{a} & -boldsymbol{b} \ mathbf{3} & mathbf{2}end{array}right] ) and ( boldsymbol{A} ) adj ( boldsymbol{A}=boldsymbol{A} boldsymbol{A}^{boldsymbol{T}} )
then ( 5 a+b ) is equal to:
A . 13
B. –
c. 5
D.
12
399 If ( boldsymbol{A}=left[begin{array}{cc}2 & 3 \ mathbf{5} & -2end{array}right] ) be such that ( boldsymbol{A}^{-1}=boldsymbol{k} boldsymbol{A} )
then find the value of ( k )
12
400 Write a ( 2 times 2 ) matrix which is both
symmetric and skew symmetric.
12
401 If ( boldsymbol{A}=left[begin{array}{cc}2 & 0 \ 5 & -3end{array}right] ) and ( B=left[begin{array}{cc}-2 & 1 \ 3 & -1end{array}right] )
then find the trace of ( left(A B^{T}right)^{T} )
12
402 Matrices obtained by changing rows and columns is called
A. rectangular matrix
B. transpose
c. symetric
D. None of the Above
12
403 A is a skew symmetric matrix such that ( A^{T} A=I, ) then ( A^{4 n-1}(n in N) ) is equal
to
A . ( -A^{T} )
B. ( I )
( c .-I )
D. ( A^{T} )
12
404 Without expanding, show that the value
of the following determinant is zero:
( left|begin{array}{ccc}mathbf{0} & boldsymbol{x} & boldsymbol{y} \ -boldsymbol{x} & boldsymbol{0} & boldsymbol{z} \ -boldsymbol{y} & -boldsymbol{z} & boldsymbol{0}end{array}right| )
12
405 Let ( A ) be a square matrix. Which of the
following is/are not skew-symmetric matrix/ces?
A . ( A-A^{T} )
B. ( A^{T}-A )
c. ( A A^{T}-A^{T} A )
D. ( A+A^{T} ), when A is skew-symmetric
12
406 ff ( left[begin{array}{cc}boldsymbol{alpha} & boldsymbol{beta} \ boldsymbol{gamma} & -boldsymbol{alpha}end{array}right] ) is square root of ( boldsymbol{I}_{2}, ) then ( boldsymbol{alpha} )
( beta ) and ( gamma ) will satisfy the relation
A. ( 1+alpha^{2}+beta gamma=0 )
В. ( 1-alpha^{2}+beta gamma=0 )
C ( cdot 1+alpha^{2}-beta gamma=0 )
D. ( -1+alpha^{2}+beta gamma=0 )
12
407 f ( a_{i j}=0(i neq j) ) and ( a_{i j}=1(i=j) )
then the matrix ( A=left[a_{i j}right]_{n times n} ) is a
matrix
A. Null
B. Identity
c. Scalar
D. Triangular
12
408 ( mathbf{f}left[begin{array}{ll}mathbf{2} & mathbf{3} \ mathbf{5} & mathbf{7}end{array}right]left[begin{array}{cc}mathbf{1} & mathbf{- 3} \ -mathbf{2} & mathbf{4}end{array}right]=left[begin{array}{ll}-mathbf{4} & mathbf{6} \ -mathbf{9} & boldsymbol{x}end{array}right], ) write
the value of ( x )
12
409 ( left|begin{array}{ccc}mathbf{1} & boldsymbol{a} & boldsymbol{a}^{2}-boldsymbol{b} boldsymbol{c} \ mathbf{1} & boldsymbol{b} & boldsymbol{b}^{2}-boldsymbol{c} boldsymbol{a} \ mathbf{1} & boldsymbol{c} & boldsymbol{c}^{2}-boldsymbol{a} boldsymbol{b}end{array}right|=? )
( mathbf{A} cdot mathbf{5} )
B. abc
( c cdot 1 )
( D )
12
410 Solve: ( left[begin{array}{ccc}mathbf{5} & mathbf{1 0} & mathbf{8} \ mathbf{3} & mathbf{2} & mathbf{6} \ mathbf{8} & boldsymbol{x}+mathbf{9} & mathbf{9}end{array}right]=mathbf{1 0 0} ) 12
411 If ( A=left[begin{array}{ccc}1 & 2 & 3 \ 2 & -3 & 0end{array}right] ) and ( B=left[begin{array}{ccc}3 & 4 & -2 \ 1 & 0 & 0end{array}right] )
then the order of ( A B^{T} ) is
( A cdot 2 times 3 )
B. ( 3 times 3 )
( c cdot 3 times 2 )
D. ( 2 times 2 )
12
412 Let ( A ) and ( B ) are two matrices of same
order ( 3 times 3 ) given by ( A= ) ( left[begin{array}{ccc}1 & 3 & lambda+2 \ 2 & 4 & 6 \ 3 & 5 & 8end{array}right] B=left[begin{array}{lll}3 & 2 & 4 \ 3 & 2 & 5 \ 2 & 1 & 4end{array}right] ) If
( t r(A B)^{t}+t r(B A)^{t}=t r(A B) ) then the
value of ( 2 lambda ) equals
A . 103
в. 206
( c .-103 )
D. -206
12
413 Find inverse of the matrix
( left[begin{array}{ccc}1 & -1 & 1 \ 0 & 1 & 1 \ 3 & 2 & -4end{array}right] ) by elementary
transformation.
12
414 If ( mathbf{5} boldsymbol{A}=left[begin{array}{cc}mathbf{3} & -mathbf{4} \ mathbf{4} & boldsymbol{x}end{array}right] ) and ( boldsymbol{A} boldsymbol{A}^{boldsymbol{T}}=boldsymbol{A}^{boldsymbol{T}} boldsymbol{A}=boldsymbol{I} )
then ( x=? )
( mathbf{A} cdot mathbf{3} )
B. -3
c. 2
D. -2
12
415 For ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} in boldsymbol{R}, ) let ( boldsymbol{A}=left[begin{array}{ccc}boldsymbol{alpha}^{2} & boldsymbol{6} & boldsymbol{8} \ boldsymbol{3} & boldsymbol{beta}^{2} & boldsymbol{9} \ boldsymbol{4} & boldsymbol{5} & boldsymbol{gamma}^{2}end{array}right] )
and ( boldsymbol{B}=left[begin{array}{ccc}mathbf{2} boldsymbol{alpha} & boldsymbol{3} & boldsymbol{5} \ boldsymbol{2} & boldsymbol{2} boldsymbol{beta} & boldsymbol{6} \ boldsymbol{1} & boldsymbol{4} & boldsymbol{2} boldsymbol{gamma}-boldsymbol{3}end{array}right] )
( boldsymbol{T}_{boldsymbol{r}}(boldsymbol{A})=boldsymbol{T}_{boldsymbol{r}}(boldsymbol{B}) ) then the value of
( left(frac{1}{alpha}+frac{1}{beta}+frac{1}{gamma}right) ) is
( boldsymbol{T}_{r}(boldsymbol{A}) ) is a Trace ( (boldsymbol{A}) ) of a matrix
( A )
B. 2
( c .3 )
D.
12
416 The number of ( A ) in ( T_{p} ) such that ( operatorname{det}(A) ) is not divisible by p is?
( A cdot 2 p^{2} )
B . ( p^{3}-5 p )
c. ( p^{3}-3 p )
D. ( p^{3}-p^{2} )
12
417 Construct a ( 2 times 3 ) matrix ( A=left[a_{i j}right] )
whose elements are given by ( boldsymbol{a}_{boldsymbol{i} j}= )
( mathbf{2}(boldsymbol{i}-boldsymbol{j}) )
В. ( left[begin{array}{ccc}0 & -2 & 4 \ 2 & 0 & -2end{array}right] )
( begin{array}{lll}text { c. } & {left[begin{array}{ccc}0 & -2 & -4 \ 2 & 0 & 2end{array}right]} \ & text { I }end{array} )
D. ( left[begin{array}{ccc}0 & -2 & 4 \ 2 & 0 & 2end{array}right] )
12
418 If ( boldsymbol{A}=left(begin{array}{ccc}1 & -1 & 3 & 2 \ 5 & -4 & 7 & 4 \ 6 & 0 & 9 & 8end{array}right), ) Write down the
elements ( a_{24} ) and ( a_{32} )
12
419 If ( A ) is an ( m times n ) matrix such that ( A B )
and ( B A ) are both defined, then order of
( B ) is
( mathbf{A} cdot m times n )
в. ( n times m )
c. ( n times n )
D. ( m times m )
12
420 If ( boldsymbol{A}=left(begin{array}{ccc}1 & 0 & 2 \ 0 & 2 & 1 \ 2 & 0 & 3end{array}right) ) and ( A^{3}-6 A^{2}+ )
( 7 A+k I^{3}=0, ) find ( k )
12
421 Assertion ( operatorname{Let} boldsymbol{A}=left[begin{array}{ll}boldsymbol{a}_{11} & boldsymbol{a}_{12} \ boldsymbol{a}_{21} & boldsymbol{a}_{22}end{array}right], boldsymbol{X}=left[begin{array}{l}boldsymbol{x}_{1} \ boldsymbol{x}_{2}end{array}right], boldsymbol{y}= )
( left[begin{array}{l}boldsymbol{y}_{1} \ boldsymbol{y}_{2}end{array}right. )
If ( A ) is symmetric, then ( X^{prime} A Y=Y^{prime} A X )
for each pair of ( X ) and ( Y )
Reason
If ( boldsymbol{X}^{prime} boldsymbol{A} boldsymbol{Y}=boldsymbol{Y}^{prime} boldsymbol{A} boldsymbol{X} ) for each pair of ( boldsymbol{X} )
and ( Y, ) then ( A ) is symmetric.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
422 Let ( p ) be a nonsingular matrix, and ( I+ )
( boldsymbol{p}+boldsymbol{p}^{2}+ldots . .+boldsymbol{p}^{n}=mathbf{0}, ) then find ( boldsymbol{p}^{-1} )
A . ( I )
B . ( p^{n+1} )
c. ( p^{n} )
D ( cdotleft(p^{n+1}-Iright)(p-I) )
12
423 For matrix ( mathbf{A} )
( (boldsymbol{alpha}+boldsymbol{beta}) boldsymbol{A}= )
A ( . alpha A )
B. ( alpha A+beta B )
( mathbf{c} cdot alpha A+beta A )
D. ( alpha^{2} A+beta^{2} A )
12
424 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{3} & mathbf{7} \ mathbf{2} & mathbf{5}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{cc}-mathbf{3} & mathbf{2} \ mathbf{4} & -mathbf{1}end{array}right] ) find
the ( boldsymbol{A}+boldsymbol{B} )
12
425 ( mathbf{f} boldsymbol{A}=left[begin{array}{lll}mathbf{0} & mathbf{1} & mathbf{2} \ mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{3} & boldsymbol{a} & mathbf{1}end{array}right] ) and ( boldsymbol{A}^{-mathbf{1}}= )
( left[begin{array}{ccc}1 / 2 & -1 / 2 & 1 / 2 \ -4 & 3 & c \ 5 / 2 & -3 / 2 & 1 / 2end{array}right] )
then
A ( a=2, c=1 / 2 )
2
В. ( a=1, c=-1 )
c. ( a=-1, c=1 )
D. ( a=1 / 2, c=1 / 2 )
12
426 If the matrix ( A ) is both symmetric and
skew symmetric, then
A. ( A ) is a diagonal matrix
B. ( A ) is a zero matrix
c. ( A ) is a square matrix
D. None of these
12
427 10. Let X and Y be two arbitrary, 3 x 3, non-zero,
ric matrices and Z be an arbitrary
bitrary, 3 x 3, non-zero, skew-symmet-
ces and Z be an arbitrary 3 x 3, non zero, symmetric
matrix. Then which of the following matrices is (ar)
symmetric?
(JEE Adv. 2015)
(a) Y24-243
(b) X44+ y44
(C) X473 -2284
(d) X23 + y23
12
428 A square matrix ( left(a_{i j}right) ) in which ( a_{i j}=0 )
for ( i neq j ) and ( a_{i j}=k(text { constant }) ) for ( i= )
( j ) is a
A. Unit matrix
B. Scalar matrix
c. Null matrix
D. none
12
429 If ( boldsymbol{A}=left[begin{array}{rr}4 & boldsymbol{x}+mathbf{2} \ mathbf{2} boldsymbol{x}-mathbf{3} & boldsymbol{x}+mathbf{1}end{array}right] ) is symmetric
then ( x= )
( A cdot 3 )
B. 5
( c cdot 2 )
D. 4
12
430 f ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{4}end{array}right] boldsymbol{B}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{5}end{array}right] ) then ( boldsymbol{A}+boldsymbol{B} ) 12
431 Simplify:
( operatorname{os} theta cdotleft[begin{array}{cc}cos theta & sin theta \ -sin theta & cos thetaend{array}right]+ )
( operatorname{in} theta cdotleft[begin{array}{cc}sin theta & -cos theta \ cos theta & sin thetaend{array}right] )
12
432 Compute ( left[begin{array}{cc}cos ^{2} x & sin ^{2} x \ sin ^{2} x & cos ^{2} xend{array}right]+ )
( left[begin{array}{cc}sin ^{2} x & cos ^{2} x \ cos ^{2} x & sin ^{2} xend{array}right] )
12
433 If ( A ) is a square matrix of order 5 and
( left.9 A^{-1}=4 A^{T} text { then ladj (adj }(operatorname{adj} A)right) )
(where ( A^{-T} ) and adj ( (A) ) denotes the inverse, transpose and adjoint of matrix A respectively) contains: ( (log 3= )
( mathbf{0 . 4 7 7}, log mathbf{2}=mathbf{0 . 3 0 3}) )
A. 56 digits
B. 60 digits
c. 58 digits
D. 53 digits
12
434 ff ( boldsymbol{A}=left[begin{array}{cc}mathbf{2} & mathbf{- 1} \ mathbf{- 1} & mathbf{2}end{array}right] ) Find ( boldsymbol{A}^{2} ) 12
435 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{2} & mathbf{3} \ mathbf{1} & -mathbf{4}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{cc}mathbf{1} & -mathbf{2} \ -mathbf{1} & mathbf{3}end{array}right] )
then verify that ( (A B)^{-1}=B^{-1} A^{-1} )
12
436 If ( A^{-1}=left[begin{array}{ccc}3 & -1 & 1 \ -15 & 6 & -5 \ 5 & -2 & 2end{array}right] ) and ( B= )
( left[begin{array}{ccc}1 & 2 & -2 \ -1 & 3 & 0 \ 0 & -2 & 1end{array}right] ), then
( (A B)^{-1}=? )
12
437 Find the inverse of the following ( operatorname{matrices} boldsymbol{A}=left(begin{array}{ccc}mathbf{2} & mathbf{1} & mathbf{3} \ mathbf{5} & mathbf{3} & mathbf{1} \ mathbf{3} & mathbf{2} & mathbf{3}end{array}right) ) 12
438 If ( boldsymbol{A}=left(begin{array}{ll}2 & 2 \ 9 & 4end{array}right) ; I=left(begin{array}{ll}1 & 0 \ 0 & 1end{array}right), ) then
( 10 A^{-1} ) is equal to
A ( .4 I-A )
B. ( 6 I-A )
c. ( A-4 I )
D. ( A-6 I )
12
439 Identify the matrix given below:
( left[begin{array}{lll}1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 2end{array}right] )
12
440 ( A=left[begin{array}{lll}1 & 0 & 0 \ 2 & 1 & 0 \ 3 & 2 & 1end{array}right], U_{1}, U_{2} ) and ( U_{3} ) are
columns matrices satisfying ( boldsymbol{A} boldsymbol{U}_{1}= ) ( left[begin{array}{l}1 \ 0 \ 0end{array}right], A U_{2}=left[begin{array}{l}2 \ 3 \ 0end{array}right], A U_{3}=left[begin{array}{l}2 \ 3 \ 1end{array}right] ) and ( U ) is
( 3 times 3 ) matrix whose columns are
( U_{1}, U_{2}, U_{3} ) then answer the following question The value of ( left[begin{array}{lll}mathbf{3} & mathbf{2} & mathbf{0}end{array}right] boldsymbol{U}left[begin{array}{l}mathbf{3} \ mathbf{2} \ mathbf{0}end{array}right] ) is
( A cdot[5] )
в. ( left[frac{5}{2}right] )
( c cdot[4] )
D. ( left[frac{3}{2}right. )
12
441 Identify the matrix given below:
( left[begin{array}{lll}1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 2end{array}right] )
A. unit matrix
B. scalar matrix
c. zero matrix
D. diagonal matrix
12
442 covert it into an upper triangular matrix Convert ( left[begin{array}{l}1-1 \ 23end{array}right] ) into an identity matrix by suitable row transformations. 12
443 If ( boldsymbol{A}=left[begin{array}{ccc}1 & -2 & 3 \ -4 & 2 & 5end{array}right] ) and ( B= )
( left[begin{array}{cc}1 & 3 \ -1 & 0 \ 2 & 4end{array}right], ) then show that ( (A B)^{prime}= )
( boldsymbol{B}^{prime} boldsymbol{A}^{prime} )
12
444 If ( A ) and ( B ) are skew symmetric
matrices of same order then
A. ( A B ) is skew symmetric
B. ( A B+B A ) is symmetic
C. ( A B-B A ) is symmetric
D. none of these
12
445 If the matrix is a square matrix and it contains 36 elements, then the order of the matrix is:
( mathbf{A} cdot 4 times 4 )
B. ( 8 times 8 )
( mathbf{c} cdot 6 times 6 )
D. ( 3 times 3 )
12
446 Find ( boldsymbol{x}+boldsymbol{y} ) if ( left[begin{array}{cc}-2 & 0 \ 3 & 1end{array}right]left[begin{array}{l}-1 \ 2 xend{array}right]+ )
( left[begin{array}{c}-2 \ 1end{array}right]=2left[begin{array}{l}y \ 3end{array}right] )
12
447 ( fleft(begin{array}{cc}a & b \ c & -aend{array}right) ) such that ( A^{2}-I ) then 12
448 fthe matrix ( A ) is such that ( left[begin{array}{ll}1 & 3 \ 0 & 1end{array}right] A= ) ( left[begin{array}{cc}mathbf{1} & mathbf{1} \ mathbf{0} & -mathbf{1}end{array}right], ) then what is equal to ( mathbf{A} ? )
A. ( left[begin{array}{cc}1 & 4 \ 0 & -1end{array}right] )
в. ( left[begin{array}{ll}1 & 4 \ 0 & 1end{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}-1 & 4 \ 0 & -1end{array}right] )
D. ( left[begin{array}{cc}1 & -4 \ 0 & -1end{array}right] )
12
449 Compute the following:
(i) ( left[begin{array}{cc}boldsymbol{a} & boldsymbol{b} \ -boldsymbol{b} & boldsymbol{a}end{array}right]+left[begin{array}{ll}boldsymbol{a} & boldsymbol{b} \ boldsymbol{b} & boldsymbol{a}end{array}right] )
(ii) ( left[begin{array}{ll}boldsymbol{a}^{2}+boldsymbol{b}^{2} & boldsymbol{b}^{2}+boldsymbol{c}^{2} \ boldsymbol{a}^{2}+boldsymbol{c}^{2} & boldsymbol{a}^{2}+boldsymbol{b}^{2}end{array}right]+ )
( left[begin{array}{cc}2 a b & 2 b c \ -2 a c & -2 a bend{array}right] )
(iii) ( left[begin{array}{cc}-14-6 \ 8516 \ 285end{array}right]+left[begin{array}{c}1276 \ 805 \ 324end{array}right] )
( (operatorname{iv})left[begin{array}{cc}cos ^{2} x & sin ^{2} x \ sin ^{2} x & cos ^{2} xend{array}right]+left[begin{array}{cc}sin ^{2} x & cos ^{2} x \ cos ^{2} x & sin ^{2} xend{array}right] )
12
450 ( mathbf{I f A}=left{begin{array}{ccc}mathbf{2} & boldsymbol{x}-mathbf{3} & boldsymbol{x}-mathbf{2} \ mathbf{3} & mathbf{- 2} & mathbf{- 1} \ mathbf{4} & mathbf{- 1} & mathbf{- 5}end{array}right} ) is a
symmetric matrix then
( A cdot 0 )
B. 3
( c .6 )
( D )
12
451 ff ( boldsymbol{A}=left[begin{array}{ll}boldsymbol{a} & boldsymbol{b} \ boldsymbol{c} & boldsymbol{d}end{array}right] ) such that ( boldsymbol{A} ) satisfies the
relation ( boldsymbol{A}^{2}-(boldsymbol{a}+boldsymbol{d}) boldsymbol{A}=mathbf{0}, ) then
inverse of ( boldsymbol{A} ) is
A . ( I )
в.
c. ( (a+d) A )
D. none of these
12
452 fthe matrix ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{4}end{array}right], ) then ( boldsymbol{I}+boldsymbol{A}+ )
( A^{2}+ldots ldots ldots . ) upto ( A^{infty}=ldots )
( mathbf{A} cdotleft[begin{array}{ll}1 & 0 \ 0 & 1end{array}right] )
В. ( left[begin{array}{cc}frac{1}{2} & frac{1}{3} \ frac{1}{-2} & 0end{array}right] )
C. ( left[begin{array}{ll}frac{-1}{2} & frac{1}{-3} \ frac{1}{-2} & 0end{array}right] )
D. ( left[begin{array}{cc}frac{1}{2} & frac{1}{-3} \ frac{1}{-2} & 0end{array}right] )
12
453 If ( boldsymbol{x}left[begin{array}{l}2 \ mathbf{3}end{array}right]+boldsymbol{y}left[begin{array}{c}-mathbf{1} \ mathbf{1}end{array}right]=left[begin{array}{c}mathbf{1 0} \ mathbf{5}end{array}right] )
Find values of ( x ) and ( y )
12
454 ( mathbf{f} mathbf{Delta}=left|begin{array}{ccc}mathbf{1} & mathbf{5} & mathbf{6} \ mathbf{0} & mathbf{1} & mathbf{7} \ mathbf{0} & mathbf{0} & mathbf{1}end{array}right| ) and ( Delta^{prime}=left|begin{array}{ccc}mathbf{1} & mathbf{0} & mathbf{1} \ mathbf{3} & mathbf{0} & mathbf{3} \ mathbf{4} & mathbf{6} & mathbf{1 0 0}end{array}right| )
then
A ( cdot Delta^{2}-3 Delta^{prime}=0 )
B. ( left(Delta+Delta^{prime}right)^{2}-3left(Delta+Delta^{prime}right)+2=0 )
c. ( left(Delta+Delta^{prime}right)^{2}+3left(Delta+Delta^{prime}right)+5=0 )
D. ( Delta+3 Delta^{prime}+1=0 )
12
455 If ( boldsymbol{A} ) is a ( mathbf{3} times mathbf{3} ) matrix ( |mathbf{3} boldsymbol{A}|=boldsymbol{k}|boldsymbol{A}| ), then
write the value of ( k )
12
456 Trace of ( A^{50} ) equals
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
12
457 Define a scalar matrix. 12
458 Find the matrix ( boldsymbol{A}=left[begin{array}{lll}mathbf{1} & mathbf{1} & mathbf{0} \ mathbf{1} & mathbf{2} & mathbf{1} \ mathbf{2} & mathbf{1} & mathbf{0}end{array}right], ) which
of the following is correct
A. ( A^{3}+3 A^{2}-I=0 )
B. ( A^{3}-3 A^{2}-I=0 )
c. ( A^{3}+2 A^{2}-I=0 )
D. ( A^{3}-A^{2}+I=0 )
12
459 A matrix ( A=left(A_{i j}right)_{m times n} ) is said to be a
square matrix if
( mathbf{A} cdot m=n )
в. ( m leq n )
c. ( m geq n )
D. ( m<n )
12
460 If ( boldsymbol{A}=left[begin{array}{ccc}1 & -2 & 3 \ -4 & 2 & 5end{array}right] ) and ( B= )
( left[begin{array}{cc}1 & 3 \ -1 & 0 \ 2 & 4end{array}right] . ) Show that ( (A B)^{prime}=? )
A ( cdot B^{prime} A^{prime} )
в. ( A^{prime} B^{prime} )
c. ( A B^{prime} )
D. ( A^{prime} B )
12
461 If ( A ) and ( B ) are square matrices of order ( n )
( x ) n such that ( A^{2}-B^{2}= )
( (A-B)(A+B), ) then of the following
will always be true?
( A cdot A=B )
c. either of A or B is a zero matrix
D. either of A or B is an identify matrix
12
462 If ( A ) is square matrix such that ( boldsymbol{A}(boldsymbol{A} boldsymbol{d} boldsymbol{j} boldsymbol{A})=left(begin{array}{ccc}mathbf{4} & mathbf{0} & mathbf{0} \ mathbf{0} & mathbf{4} & mathbf{0} \ mathbf{0} & mathbf{0} & mathbf{4}end{array}right) ) then det ( (mathbf{A} mathrm{d} )
( A)= )
( A cdot 4 )
B. 16
( c cdot 64 )
D. 256
12
463 ( $ $ text { lbegin{array }}{mid} A=mid ) left [ lbegin{array ( }[1332-4115 & 2 ) ( mathrm{~ l e n d { a r r a y } ~ | r i g h t ] ~ | , ~ | , ~ B = | l e f t [ ~} )
[
begin{array}{cc}
1 & -3 \
5 & 8
end{array}
]
|right] ( |=3 ) left Then find ( 3 A-5 B+ )
( 4 I ) by using matrix
12
464 Find ( X, ) if ( Y=left[begin{array}{ll}3 & 2 \ 1 & 4end{array}right] ) and ( 2 X+Y= )
( left[begin{array}{cc}1 & 0 \ -3 & 2end{array}right] )
12
465 ff ( boldsymbol{A}=left(begin{array}{cc}mathbf{3} & mathbf{1} \ -mathbf{9} & -mathbf{3}end{array}right) ) then
( left(1+2 A+3 A^{2}+ldots . inftyright)^{-1} ) equals
( ^{A} cdotleft(begin{array}{cc}-5 & -2 \ 18 & 7end{array}right) )
B. ( left(begin{array}{rr}-5 & 18 \ -2 & 7end{array}right) )
c. ( left(begin{array}{cc}7 & -2 \ 18 & -5end{array}right) )
D. None of these
12
466 If ( A ) is skew-symmetric, then ( A^{n} ) for
( boldsymbol{n} in boldsymbol{N} ) is
This question has multiple correct options
A. Symmetric
B. Skew-symmetric
c. Diagonal
D. None of these
12
467 Construct a ( 3 times 2 ) matrix ( A=left[a_{i j}right] ) whose elements are given by ( a_{i j}=frac{i}{j} ) 12
468 ( boldsymbol{A}=left[begin{array}{rrr}mathbf{1} & mathbf{- 2} & mathbf{3} \ mathbf{7} & -mathbf{8} & mathbf{9} \ mathbf{4} & mathbf{- 5} & mathbf{6}end{array}right] ) the new matrix
formed by adding ( 2^{n d} ) row to ( 1^{s t} ) row will be
A. ( left[begin{array}{ccc}8 & -10 & 12 \ 7 & -8 & 9 \ 4 & -5 & 6end{array}right] )
В. ( left[begin{array}{lll}6 & 6 & 6 \ 7 & 8 & 9 \ 4 & 5 & 6end{array}right] )
c. ( left[begin{array}{ccc}1 & 2 & 3 \ 7 & 8 & 9 \ 11 & -13 & 14end{array}right] )
D. ( left[begin{array}{ccc}1 & -2 & 3 \ 7 & 8 & -29 \ 4 & -2 & 6end{array}right] )
12
469 By row transformation find ( boldsymbol{A}^{-1} ) if:
( boldsymbol{A}=left[begin{array}{ll}mathbf{2} & mathbf{1} \ mathbf{4} & mathbf{2}end{array}right] )
12
470 ( mathbf{f} boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{2} & mathbf{- 3} \ mathbf{5} & mathbf{0} & mathbf{2} \ mathbf{1} & mathbf{- 1} & mathbf{1}end{array}right], boldsymbol{B}= )
( left[begin{array}{ccc}mathbf{3} & mathbf{- 1} & mathbf{2} \ mathbf{4} & mathbf{2} & mathbf{5} \ mathbf{2} & mathbf{0} & mathbf{3}end{array}right], boldsymbol{C}=left[begin{array}{ccc}mathbf{4} & mathbf{1} & mathbf{2} \ mathbf{0} & mathbf{3} & mathbf{2} \ mathbf{1} & mathbf{- 2} & mathbf{3}end{array}right], ) Then
Compute ( (boldsymbol{A}+boldsymbol{B}) ) and ( (boldsymbol{B}-boldsymbol{C}) . ) Also
verify that ( boldsymbol{A}+(boldsymbol{B}-boldsymbol{C})=(boldsymbol{A}+boldsymbol{B})- )
( C )
12
471 If ( A B=0, ) then for the matrices ( A= ) ( left[begin{array}{cc}cos ^{2} theta & cos theta sin theta \ cos theta sin theta & sin ^{2} thetaend{array}right] ) and ( B= )
( left[begin{array}{cc}cos ^{2} phi & cos phi sin phi \ cos phi sin phi & sin ^{2} phiend{array}right], theta-phi ) is
A ( cdot ) an odd muliple of ( frac{pi}{2} )
B. an odd multiple of ( pi )
C . an even multiple of ( frac{pi}{2} )
D.
12
472 Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] ) whose element ( a_{i j} ) is ( a_{i j}=frac{(i-2 j)^{2}}{2} ) 12
473 Find the matrices ( A ) and ( B ) such that ( boldsymbol{A}+boldsymbol{B}=left[begin{array}{ll}mathbf{5} & mathbf{4} \ mathbf{7} & mathbf{3}end{array}right] ) and ( boldsymbol{A}-boldsymbol{B}= )
( left[begin{array}{cc}11 & 2 \ -1 & 7end{array}right] )
12
474 If ( A ) is a ( 2 times 3 ) matrix and ( B ) is ( 3 times 2 )
matrix then the order of ( (A B)^{T} ) is equal
to the order of
( mathbf{A} cdot A B )
в. ( A^{T} B^{T} )
( c . ) ВА
D. All of these
12
475 ff ( boldsymbol{A}=left[begin{array}{cc}boldsymbol{6} & boldsymbol{2} \ boldsymbol{5} & boldsymbol{-} boldsymbol{4}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{cc}mathbf{1} & boldsymbol{2} \ -boldsymbol{5} & boldsymbol{1}end{array}right], ) find
a matrix ( X ) such that ( 2 A+3 B- )
( mathbf{5} boldsymbol{X}=mathbf{0} )
12
476 ( operatorname{Let} A=left[begin{array}{lll}1 & 0 & 0 \ 1 & 1 & 0 \ 1 & 1 & 1end{array}right] ) and ( B=A^{10} )
Then the sum of elements of the first
column of ( boldsymbol{B} ) is
12
477 3.
Let M and N be two 3 x 3 non-singular skew- symmetric
matrices such that MN = NM. If PT denotes the transpose
of P, then M²N2 (MTN)-1 (MN-1)T is equal to (2011)
(a) M2 (b) N2 (C) – M2 (d) MN
12
478 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{2} & -mathbf{3} \ mathbf{4} & mathbf{1}end{array}right], boldsymbol{B}=left[begin{array}{cc}mathbf{2} & mathbf{3} \ mathbf{5} & mathbf{0}end{array}right] ) and
( boldsymbol{C}=left[begin{array}{cc}-mathbf{1} & mathbf{2} \ mathbf{0} & mathbf{5}end{array}right], ) then find ( boldsymbol{A}(boldsymbol{B}+boldsymbol{C}) )
12
479 What is meant by transposing of a
matrix? Give an example.
12
480 Solve for ( x ) and
( boldsymbol{y} )
( mathbf{2}left[begin{array}{cc}boldsymbol{x} & mathbf{7} \ mathbf{9} & boldsymbol{y}-mathbf{5}end{array}right]+left[begin{array}{cc}mathbf{6} & -mathbf{7} \ mathbf{4} & mathbf{5}end{array}right]=left[begin{array}{cc}mathbf{1 0} & mathbf{7} \ mathbf{2 2} & mathbf{1 5}end{array}right] )
12
481 If ( boldsymbol{A}^{T}=left[begin{array}{cc}mathbf{4} & mathbf{5} \ -mathbf{1} & mathbf{0} \ mathbf{2} & mathbf{3}end{array}right] ) and ( boldsymbol{B}= )
( left[begin{array}{ccc}2 & -1 & 1 \ 7 & 5 & -2end{array}right], ) verify the following
( mathbf{A} cdot(A+B)^{T}=A^{T}+B^{T}=B^{T}+A^{T} )
В ( cdot(A+B)^{T}=A^{T}-B^{T} )
c. ( left(B^{T}right)^{T}=B )
D. none of these
12
482 ff ( boldsymbol{A}=left[begin{array}{cc}cos boldsymbol{x} & sin boldsymbol{x} \ -sin boldsymbol{x} & cos boldsymbol{x}end{array}right], ) then find ( boldsymbol{x} )
satisfying ( mathbf{0}<boldsymbol{x}<frac{boldsymbol{pi}}{mathbf{2}} ) when ( boldsymbol{A}+boldsymbol{A}^{boldsymbol{T}}=boldsymbol{I} )
12
483 ( mathrm{IF} mathrm{A}=left|begin{array}{ll}mathbf{1} & mathbf{0} \ mathbf{1} & mathbf{0}end{array}right| ) And ( mathrm{B}=left|begin{array}{ll}mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{1}end{array}right| ) then ( mathbf{A}+mathbf{B}= )
( A cdot A )
B. B
c. ( mid begin{array}{ll}2 & 0 \ 1 & 1end{array} )
D. ( left|begin{array}{ll}0 & 2 \ 2 & 2end{array}right| )
12
484 If ( boldsymbol{A}=left[begin{array}{ccc}1 & 1 & -1 \ 2 & -3 & 4 \ 3 & -2 & 3end{array}right] ) and ( B= )
( left[begin{array}{ccc}-1 & -2 & -1 \ 6 & 12 & 6 \ 5 & 10 & 5end{array}right], ) then which of the
following is/are correct?
1. ( A ) and ( B ) commute.
2. AB is null matrix.
Select the correct answer using the
code given below:
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2
12
485 If ( A ) is a matrix of order ( 3 times 4 ), then both
( A B^{T} ) and ( B^{T} A ) are defined if order of ( B )
is
( A cdot 3 times 3 )
B. ( 4 times 4 )
c. ( 4 times 3 )
D. ( 3 times 4 )
12
486 ( mathrm{f} A=left[begin{array}{lll}1 & 3 & 3 \ 1 & 4 & 3 \ 1 & 3 & 4end{array}right] ) then find ( A^{-1} ) 12
487 Using elementary tansormations, find
the inverse of each of the matrices, if it
exists in ( left[begin{array}{ll}2 & 1 \ 7 & 4end{array}right] )
12
488 Let ( A ) be a matrix of order ( 3 times 4 . ) If ( R_{1} )
denotes the first row of ( A ) and ( C_{2} )
denotes its second column, then
determine the orders of matrices ( boldsymbol{R}_{1} )
and ( C_{2} )
12
489 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{3} & mathbf{1} \ -mathbf{1} & mathbf{2} \ mathbf{0} & mathbf{6}end{array}right] ) and ( boldsymbol{B}= )
( left[begin{array}{ccc}5 & 4 & 6 \ 4 & 1 & 2 \ -5 & -1 & 1end{array}right], ) then
A. ( A+B ) exists
B. ( A B ) exists
c. ( B A ) exists
D. none of these
12
490 If ( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{- 2} & mathbf{3} \ -mathbf{4} & mathbf{2} & mathbf{5}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{cc}mathbf{2} & mathbf{3} \ mathbf{4} & mathbf{5} \ mathbf{2} & mathbf{1}end{array}right] )
Check commutativity of the two matrices
12
491 not the square of a 3 x 3
(JEE Adv. 2017)
a=-3.
13. Which of the following is(are) not the squ
matrix with real entries?
(1 o o7
To ol
(a) Tolol
Tolo
Lo 0 1
(b)
Lo 0 -1]
1007
Hool
(c) 0 -1 0
(d) 1 0 -1 0
Lo o 1
To 0 -1
17
12
492 If ( boldsymbol{a}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{4} & mathbf{2}end{array}right], ) than show that ( |mathbf{2} boldsymbol{A}|= )
( mathbf{4}|boldsymbol{A}| )
12
493 Convert ( left[begin{array}{cc}1 & -1 \ 2 & 3end{array}right] ) into on identity matrix by suitable raw transformation. 12
494 Let ( A ) be a symmetric matrix such that
( boldsymbol{A}^{5}=mathbf{0} ) and ( boldsymbol{B}=boldsymbol{I}+boldsymbol{A}+boldsymbol{A}^{2}+boldsymbol{A}^{3}+boldsymbol{A}^{4} )
then ( B ) is
This question has multiple correct options
A. symmetric
B. singular
c. non-singular
D. skew symmetric
12
495 Matrix ( boldsymbol{A}=left[boldsymbol{a}_{i j}right]_{m times n} ) is a square matrix
if
( A cdot mn )
( c cdot m=1 )
( D cdot m=n )
12
496 Using elementary transformation, find the inverse of the matrix ( left[begin{array}{ccc}2 & -3 & 3 \ 2 & 2 & 3 \ 3 & -2 & 2end{array}right] ) 12
497 Let ( A=left(begin{array}{ll}1 & 2 \ 3 & 4end{array}right) ) and ( B= )
( left(begin{array}{ll}a & 0 \ 0 & bend{array}right), a, b in N . ) Then:
A. there exists exactly one B such that ( A B=B A )
B. there exist exactly infinitely many B’s such that ( A B= ) ( B A )
C. there cannot exist any B such that ( A B=B A )
D. there exist more than one but finite number of B’s such that ( A B=B A )
12
498 If ( A=left|begin{array}{cc}0 & 1 \ 2 & 4end{array}right|, B=left|begin{array}{cc}-1 & 1 \ 2 & 2end{array}right| )
( c=left|begin{array}{cc}1 & 0 \ 1 & 0end{array}right|, ) then ( 2 A+3 B-C= )
A ( cdotleft|begin{array}{cc}-4 & 5 \ 9 & 14end{array}right| )
в. ( left|begin{array}{cc}4 & 3 \ 9 & 10end{array}right| )
с. ( left|begin{array}{cc}4 & -5 \ 9 & 14end{array}right| )
О ( cdotleft|begin{array}{cc}-4 & 5 \ 14 & 9end{array}right| )
12
499 The order the matrix is ( left[begin{array}{lll}2 & 3 & 4 \ 9 & 8 & 7end{array}right] ) is
( mathbf{A} cdot 4 times 3 )
B. ( 3 times 2 )
( c cdot 2 times 3 )
D. ( 3 times 1 )
12
500 ( left[begin{array}{lll}mathbf{1} & mathbf{1} & boldsymbol{x}end{array}right]left[begin{array}{lll}mathbf{1} & mathbf{0} & mathbf{2} \ mathbf{0} & mathbf{2} & mathbf{1} \ mathbf{2} & mathbf{1} & mathbf{0}end{array}right]left[begin{array}{l}mathbf{1} \ mathbf{1} \ mathbf{1}end{array}right]=mathbf{0}, ) then
find ( x )
12
501 Inverse of a diagonal non-singular matrix is
A. Scalar matrix
B. Skew symmetric matrix
c. zero matrix
D. Diagonal matrix
12
502 If ( boldsymbol{m}left[begin{array}{ll}-mathbf{3} & mathbf{4}end{array}right]+boldsymbol{n}left[begin{array}{ll}mathbf{4} & -mathbf{3}end{array}right]=left[begin{array}{ll}mathbf{1 0} & -mathbf{1 1}end{array}right] )
then ( 3 m+7 n= )
( A cdot 3 )
B. 5
c. 10
( D )
12
503 Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] )
whose element ( a_{i j} ) is ( a_{i j}=frac{(i-j)^{2}}{2} )
12
504 ( boldsymbol{A}=left[begin{array}{cc}-mathbf{3} & -mathbf{5} \ -mathbf{6} & mathbf{0}end{array}right], boldsymbol{A}-boldsymbol{B}=mathbf{2} boldsymbol{I} . ) Find ( boldsymbol{B} )
( mathbf{A} cdotleft[begin{array}{cc}-5 & -5 \ -6 & -2end{array}right] )
в. ( left[begin{array}{cc}1 & 2 \ -3 & -6end{array}right] )
с. ( left[begin{array}{cc}5 & 5 \ -3 & -6end{array}right] )
О ( cdotleft[begin{array}{ll}1 & 2 \ 3 & 6end{array}right] )
12
505 If ( A ) is ( 3 times 4 ) matrix and ( B ) is a matrix
such that ( A^{prime} B ) and ( B^{prime} A ) are both
defined, then the order of ( B ) is
( mathbf{A} cdot 4 times 4 )
B. ( 3 times 3 )
c. ( 3 times 4 )
D. ( 4 times 3 )
12
506 Find the inverse of the following matrix by using elementary row transformation
( left[begin{array}{ll}2 & 5 \ 1 & 3end{array}right] )
12
507 ff ( left[begin{array}{cc}1 & 2 \ 3 & -5end{array}right], ) then ( A^{-1} ) is equal to
( ^{mathbf{A}} cdotleft[begin{array}{cc}frac{5}{11} & frac{2}{11} \ frac{3}{11} & -frac{1}{11}end{array}right] )
в. ( left[begin{array}{rr}-frac{5}{11} & -frac{2}{11} \ -frac{3}{11} & -frac{1}{11}end{array}right] )
( ^{mathbf{C}} cdotleft[begin{array}{cc}frac{5}{11} & frac{2}{11} \ frac{3}{11} & frac{1}{11}end{array}right] )
D. ( left[begin{array}{ll}5 & 2 \ 3 & -1end{array}right] )
12
508 25. Let A and B be two symmetric matrices of order 3.
Statement-1: A(BA) and (AB)A are symmetric matrices
Statement-2: AB is symmetric matrix ifmatrix multiplicat
of A with B is commutative.
[2011]
(a) Statement-1 is true, Statement-2 is true; Statement-
not a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is false.
(C) Statement-1 is false, Statement-2 is true.
(d) Statement-1 is true, Statement-2 is true; Statement-2 is
a correct explanation for Statement-1.
12
509 If ( A ) and ( B ) are ( 3 times 3 ) matrices and
( |boldsymbol{A}| neq mathbf{0}, ) then
This question has multiple correct options
A ( cdot|A B|=0 Rightarrow|B|=0 )
B . ( |A B| neq 0 Rightarrow|B| neq 0 )
C ( cdotleft|A^{-1}right|=|A|^{-1} )
D. ( |2 A|=2|A| )
12
510 If ( Delta_{r}=left|begin{array}{ccc}r-1 & n & 6 \ (r-1)^{2} & 2 n^{2} & 4 n-2 \ (r-1)^{3} & 3 n^{3} & 3 n^{2}-3 nend{array}right| ) 12
511 ( operatorname{Let} boldsymbol{A}=left[begin{array}{ll}mathbf{0} & boldsymbol{a} \ mathbf{0} & mathbf{0}end{array}right] ) and ( (boldsymbol{A}+boldsymbol{I})^{50} )
( mathbf{5} mathbf{0} boldsymbol{A}=left[begin{array}{ll}boldsymbol{a} & boldsymbol{b} \ boldsymbol{c} & boldsymbol{d}end{array}right] . ) Then the value of ( boldsymbol{a}+ )
( boldsymbol{b}+boldsymbol{c}+boldsymbol{d} ) is
A . 2
B.
( c cdot 4 )
D. none of these
12
512 ff ( Delta=left|begin{array}{lll}a r g z_{1} & a r g z_{2} & a r g z_{3} \ a r g z_{2} & a r g z_{3} & a r g z_{1} \ a r g z_{3} & a r g z_{1} & a r g z_{2}end{array}right|, ) the ( , Delta )
is divided by:
A ( cdot arg left(z_{1}+z_{2}+z_{3}right) )
B ( cdot arg left(z_{1} cdot z_{2} cdot z_{3}right) )
C ( cdotleft(a r g z_{1}+a r g z_{2}+arg z_{3}right) )
D. N.O.T
12
513 ( operatorname{Let} boldsymbol{A}=left[begin{array}{cc}sin boldsymbol{theta} & mathbf{0} \ mathbf{0} & -boldsymbol{operatorname { s i n } boldsymbol { theta }}end{array}right] cdot ). If ( boldsymbol{A}+boldsymbol{A}^{boldsymbol{T}} ) is a
null matrix, then the number of values
of ( boldsymbol{theta} ) in ( [mathbf{0}, mathbf{2} boldsymbol{pi}] ) is
A . 4
B. 3
( c cdot 2 )
D.
12
514 If ( boldsymbol{A}=left[begin{array}{cc}cos boldsymbol{theta} & -sin boldsymbol{theta} \ sin boldsymbol{theta} & cos boldsymbol{theta}end{array}right], ) then ( boldsymbol{A} boldsymbol{A}^{boldsymbol{T}} )
equals
( A cdotleft[begin{array}{cc}cos 2 theta & -sin 2 theta \ sin 2 theta & cos 2 thetaend{array}right] )
B. ( left[begin{array}{cc}cos ^{2} theta & sin ^{2} theta \ sin ^{2} theta & cos ^{2} thetaend{array}right] )
c. ( left[begin{array}{ll}1 & 0 \ 0 & 1end{array}right] )
D. ( left[begin{array}{ll}0 & 0 \ 0 & 0end{array}right] )
12
515 If ( boldsymbol{A}=left[boldsymbol{a}_{i j}right] ) is a ( 2 times 2 ) matrix such that
( a_{i j}=i+2 j, ) then find ( A )
12
516 ( left[begin{array}{cc}boldsymbol{a}+boldsymbol{b} & boldsymbol{2} \ mathbf{5} & boldsymbol{b}end{array}right]=left[begin{array}{ll}mathbf{6} & mathbf{5} \ mathbf{2} & mathbf{2}end{array}right], ) then find ( boldsymbol{a} ) 12
517 If matrix ( boldsymbol{A}=[mathbf{1} mathbf{2} mathbf{3}], ) then find ( boldsymbol{A} boldsymbol{A}^{boldsymbol{T}} ) 12
518 Given ( boldsymbol{x}-boldsymbol{y}+mathbf{3} boldsymbol{z}=mathbf{5} ; mathbf{4} boldsymbol{x}+mathbf{2} boldsymbol{y}-boldsymbol{z}=mathbf{0} )
( ;-boldsymbol{x}+mathbf{3} boldsymbol{y}+boldsymbol{z}=mathbf{5} )
If
( mathbf{A}=left[begin{array}{ccc}mathbf{1} & mathbf{- 1} & mathbf{3} \ mathbf{4} & mathbf{2} & -mathbf{1} \ -mathbf{1} & mathbf{3} & mathbf{1}end{array}right], boldsymbol{X}=left[begin{array}{l}boldsymbol{x} \ boldsymbol{y} \ boldsymbol{z}end{array}right], boldsymbol{D}= )
( left.begin{array}{l}mathbf{5} \ mathbf{0} \ mathbf{5}end{array}right} ) such that ( mathbf{A X}=mathbf{D} )
Show that ( A ) is non singular and the
cofactor elements of a matrix ( boldsymbol{A} ) is
( left[begin{array}{ccc}+(2+3) & -(4-1) & +(12+2) \ -(-1-9) & +(1+3) & -(3-1) \ +(1-6) & -(-1-12) & +(2+4)end{array}right) )
12
519 If ( 2 A+B=left[begin{array}{cc}3 & -1 \ 2 & 4end{array}right] ) and ( B= )
( left[begin{array}{cc}-1 & -5 \ 0 & 2end{array}right], ) then find ( A )
12
520 ( operatorname{Given} mathbf{3}left[begin{array}{cc}boldsymbol{x} & boldsymbol{y} \ boldsymbol{z} & boldsymbol{w}end{array}right]=left[begin{array}{cc}boldsymbol{x} & boldsymbol{6} \ -mathbf{1} & boldsymbol{2} boldsymbol{w}end{array}right]+ )
( left[begin{array}{cc}4 & x+y \ z+w & 3end{array}right], ) find the values of
( boldsymbol{x}, boldsymbol{y}, boldsymbol{z} ) and ( boldsymbol{w} )
12
521 If ( A=left(begin{array}{ccc}1 & -1 & 3 \ 5 & -4 & 7 \ 6 & 0 & 9 & 8end{array}right), ) Find the order of the
matrix
12
522 Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] ) 12
523 If
( left(begin{array}{cc}1 & -tan theta \ tan theta & 1end{array}right)left(begin{array}{cc}1 & tan theta \ -tan theta & 1end{array}right)^{-1}= )
( left[begin{array}{cc}boldsymbol{a} & -boldsymbol{b} \ boldsymbol{b} & boldsymbol{a}end{array}right], ) then
This question has multiple correct options
( mathbf{A} cdot a=cos 2 theta )
В . ( a=1 )
c. ( b=sin 2 theta )
D. ( b=-1 )
12
524 Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] )
whose element ( boldsymbol{a}_{boldsymbol{i} j} ) is ( boldsymbol{a}_{boldsymbol{i} j}=frac{|mathbf{2} boldsymbol{i}-boldsymbol{3} boldsymbol{j}|}{boldsymbol{2}} )
12
525 ( A ) and ( B ) are symmetric matrices of the same order. ( boldsymbol{X}=boldsymbol{A B}+boldsymbol{B A} ) and ( boldsymbol{Y}= )
( boldsymbol{A B}-boldsymbol{B A} )
( (boldsymbol{X} boldsymbol{Y})^{boldsymbol{T}}= )
A . ( X Y )
в. ( Y X )
( c cdot-Y X )
D. ( X+Y )
12
526 Select the missing number from the
given matrix:
( begin{array}{ccc}5 & 2 & 4 \ 4 & 4 & 7 \ 2 & 5 & 3 \ 18 & 30 & ?end{array} )
A . 43
B. 42
( c .33 )
D. 32
12
527 Prove that:
[
begin{array}{l}
{left[begin{array}{lll}
mathbf{x} mathbf{y} & mathbf{z}
end{array}right]left[begin{array}{lll}
boldsymbol{a} & boldsymbol{h} & boldsymbol{g} \
boldsymbol{h} & boldsymbol{b} & boldsymbol{f} \
boldsymbol{g} & boldsymbol{f} & boldsymbol{c}
end{array}right]left[begin{array}{l}
boldsymbol{x} \
boldsymbol{y} \
boldsymbol{z}
end{array}right]} \
= & {left[boldsymbol{a x}^{2}+boldsymbol{b} boldsymbol{y}^{2}+boldsymbol{c} boldsymbol{z}^{2}+boldsymbol{c f} boldsymbol{z}+boldsymbol{2 g} boldsymbol{z} boldsymbol{x}+boldsymbol{t}right.}
end{array}
]
( 2 h x y )
12
528 If ( boldsymbol{A}=left[boldsymbol{a}_{i j}right] ) is a scalar matrix of order
( boldsymbol{n} times boldsymbol{n} ) such that ( boldsymbol{a}_{boldsymbol{i} j}=boldsymbol{k} ) for all then
trace of ( A ) is equal to
( A cdot n k )
в. ( n+k )
c. ( n / k )
D. none of these
12
529 f ( boldsymbol{A}=left(begin{array}{ll}1 & -1 \ 2 & -2end{array}right) ) Find ( 5 boldsymbol{I}-8 boldsymbol{A} ) 12
530 ( (A B)^{-1}= )
( A cdot B A )
B ( cdot A^{-1} B^{-1} )
c. ( B^{-1} A^{-1} )
D. All of these
12
531 fthe matrix ( left(begin{array}{cc}mathbf{6} & -boldsymbol{x}^{2} \ mathbf{2} boldsymbol{x}-mathbf{1 5} & mathbf{1 0}end{array}right) )
symmetric, find the value of ( x )
12
532 Find ( frac{1}{2}left(A+A^{T}right) ) and ( frac{1}{2}left(A-A^{T}right) )
when ( A=left[begin{array}{ccc}0 & a b \ -a & 0 & c \ -b-c 0end{array}right] )
12
533 If ( A ) is a square of order 3 , then
( left|boldsymbol{A} boldsymbol{d} boldsymbol{j}left(boldsymbol{A} boldsymbol{d} boldsymbol{j} boldsymbol{A}^{2}right)right|= )
( mathbf{A} cdot|A|^{2} )
B . ( |A|^{4} )
c. ( |A|^{8} )
D・ ( |A|^{1} )
12
534 If ( A ) is a square matrix, ( B ) is a singular
matrix of same order, then for a positive integer ( n,left(A^{-1} B Aright)^{n} ) equals
( mathbf{A} cdot A^{-n} B^{n} A^{n} )
B ( cdot A^{n} B^{n} A^{-n} )
c. ( A^{-1} B^{n} A )
D. ( nleft(A^{-1} B Aright) )
12
535 If ( A ) is square matrix of order ( 3, ) then
( left|A d jleft(A d j A^{2}right)right|= )
A ( cdot|A|^{2} )
B ( cdot|A|^{4} )
c. ( |A|^{8} )
D・ ( |A|^{16} )
12
536 ( mathbf{1} )
( left[begin{array}{ll}1 & 1 \ 0 & 1end{array}right]left[begin{array}{ll}1 & 2 \ 0 & 1end{array}right]left[begin{array}{ll}1 & 3 \ 0 & 1end{array}right] cdotleft[begin{array}{cc}1 & n-1 \ 0 & 1end{array}right]= )
( left[begin{array}{ll}1 & 78 \ 0 & 1end{array}right], ) then the inverse of ( left[begin{array}{ll}1 & n \ 0 & 1end{array}right] ) is?
A. ( left[begin{array}{cc}1 & -13 \ 0 & 1end{array}right] )
в. ( left[begin{array}{ll}1 & 0 \ 12 & 1end{array}right] )
c. ( left[begin{array}{cc}1 & -12 \ 0 & 1end{array}right] )
D. ( left[begin{array}{ll}1 & 0 \ 13 & 1end{array}right] )
12
537 ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{8} \ mathbf{9} & mathbf{4}end{array}right] boldsymbol{B}=left[begin{array}{ll}mathbf{6} & mathbf{7} \ mathbf{3} & mathbf{5}end{array}right] ) Find ( boldsymbol{A}+boldsymbol{B} ) 12
538 If a matrix has equal number of
columns and rows then it is said to be a
A. row matrix
B. identical matrix
c. square matrix
D. rectangular matrix
12
539 Evaluate
( left[begin{array}{l}3 \ 4 \ 1end{array}right]left[begin{array}{lll}2 & -1 & 3end{array}right] )
12
540 The inverse of a symmetric matrix is
A. symmetric
B. skew-symmetric
c. diagonal matrix
D. singular matrix
12
541 Show that matrix ( A+B ) is symmetric
or skew symmetric according as ( A ) and
( B ) are symmetric of skew symmetric.
12
542 ( A ) is a ( 3 times 3 ) diagonal matrix having integral entries such that ( operatorname{det}(A)=120 )
number of such matrices is ( 10 n ), then
( boldsymbol{n} ) is
A . 36
B . 38
( c cdot 34 )
D. 30
12
543 Identify a matrix
В. ( A={1,2} )
( mathbf{c} cdot A=left[begin{array}{ll}1 & 2end{array}right] )
D. None of these
12
544 Let ( boldsymbol{alpha}=boldsymbol{pi} / mathbf{5} ) and
[
begin{array}{c}
boldsymbol{A}=left[begin{array}{cc}
cos boldsymbol{alpha} & sin boldsymbol{alpha} \
-sin boldsymbol{alpha} & cos boldsymbol{alpha}
end{array}right] text { and } boldsymbol{B}=boldsymbol{A}+ \
boldsymbol{A}^{2}+boldsymbol{A}^{3}+boldsymbol{A}^{4}, text { then }
end{array}
]
This question has multiple correct options
A. singular
B. non-singular
c. skew-symmetric
D. ( |B|=1 )
12
545 Ti 007
p=
4
4
21. Let P-
1
0
and I be the identity matrix of order 3.
16
4
1
If O= [9] is a matrix such that P50 – Q =I, then
931+ 932
921
equals
(a) 52
(6) 103
(c) 201
(JEE Adv. 2016)
(2) 205
How many 3 x 3 matrico
,
12
546 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{3} & -mathbf{2} \ mathbf{4} & -mathbf{2}end{array}right], ) find ( boldsymbol{K} ) such that ( boldsymbol{A}^{mathbf{2}}= )
( boldsymbol{K} boldsymbol{A}-boldsymbol{2} boldsymbol{I}, ) where ( boldsymbol{I} ) is the identity
element.
12
547 If ( A ) is a square matrix, then ( A-A^{T} ) is
A. unit matrix
B. null matrix
( c . A )
D. a skew symmetric matrix
12
548 If ( A ) is a ( 3 times 3 ) skew-symmetric matrix,
then the trace of ( A ) is equal to
A . -1
B.
c. ( |A| )
D. 0
12
549 If ( left(A+B^{T}right)^{T} ) is a matrix of order ( 4 times 3 )
then the order of matrix B is
( A cdot 3 times 4 )
B. ( 4 times 3 )
( c cdot 3 times 3 )
D. ( 4 times 4 )
12
550 If ( boldsymbol{A}=left[begin{array}{cc}1 & tan x \ -tan x & 1end{array}right], ) then ( A^{T} A^{-1} ) is
( mathbf{A} cdotleft[begin{array}{ll}-cos 2 x & sin 2 x \ -sin 2 x & cos 2 xend{array}right] )
В. ( left[begin{array}{cc}cos 2 x & -sin 2 x \ sin 2 x & cos 2 xend{array}right] )
c. ( left[begin{array}{cc}cos 2 x & cos 2 x \ cos 2 x & sin 2 xend{array}right] )
D. none of these
12
551 f ( A+2 B=left[begin{array}{cc}2 & -4 \ 1 & 6end{array}right], A^{prime}+B^{prime}= )
( left[begin{array}{cc}1 & 2 \ 0 & -1end{array}right], ) then ( A= )
12
552 If ( boldsymbol{A}=left[begin{array}{ll}boldsymbol{i} & mathbf{0} \ mathbf{0} & boldsymbol{i}end{array}right], boldsymbol{n} in boldsymbol{N}, ) then ( boldsymbol{A}^{4 n} ) equals
( A cdotleft[begin{array}{ll}1 & 0 \ 0 & 1end{array}right] )
в. ( left[begin{array}{ll}i & 0 \ 0 & iend{array}right] )
c. ( left[begin{array}{ll}0 & i \ i & 0end{array}right] )
D. ( left[begin{array}{ll}0 & 0 \ 0 & 0end{array}right] )
12
553 Given, matrix ( boldsymbol{A}=left[begin{array}{l}mathbf{3} \ mathbf{2}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{l}-mathbf{2} \ -mathbf{1}end{array}right] )
find the matrix ( X ) such that ( X-A=B )
( A cdotleft[begin{array}{l}0 \ 0end{array}right] )
в. ( left[begin{array}{l}1 \ 1end{array}right] )
( c cdotleft[begin{array}{l}4 \ 0end{array}right] )
D. ( left[begin{array}{l}1 \ -1end{array}right. )
12
554 Find the values of ( x, y, a ) and ( b ) if ( left[begin{array}{cccc}3 x+4 y & 2 & x-2 y \ a+b & 2 a-b & -1end{array}right]= ) 12
555 For ( mathbf{3} times mathbf{3} ) matrices ( boldsymbol{A} ) and ( boldsymbol{B}, ) if ( |boldsymbol{B}|=mathbf{1} )
and ( A=2 B ) then find ( |A| )
A .
B. 4
( c cdot 2 )
D. 8
12
556 ( mathbf{f} boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{4}end{array}right], ) find ( |mathbf{2} boldsymbol{A}| ) 12
557 The construction of ( 3 times 4 ) matrix ( A )
whose elements ( a_{i j} ) is given by ( frac{(i+j)^{2}}{2} ) is
( mathbf{A} cdotleft[begin{array}{llll}2 & 9 / 2 & 8 & 25 \ 9 & 4 & 5 & 18 \ 8 & 25 & 18 & 49end{array}right] )
( mathbf{B} cdotleft[begin{array}{cccc}2 & 9 / 2 & 25 / 2 & 9 \ 9 / 2 & 5 / 2 & 5 & 45 / 2 \ 25 & 18 & 25 & 9 / 2end{array}right] )
( mathbf{C} cdotleft[begin{array}{cccc}2 & 9 / 2 & 8 & 25 / 2 \ 9 / 2 & 8 & 25 / 2 & 18 \ 8 & 25 / 2 & 18 & 49 / 2end{array}right] )
D. None of these
12
558 ( left(left[begin{array}{ll}mathbf{8} & mathbf{4} \ boldsymbol{x} & mathbf{8}end{array}right]right)=mathbf{4}left(left[begin{array}{ll}mathbf{2} & mathbf{1} \ mathbf{1} & mathbf{2}end{array}right]right) ) then the value
of ( x ) is
A .
B. 2
( c cdot frac{1}{4} )
( D )
12
559 Assertion ( operatorname{Let} a, b in R, ) and ( I=left[begin{array}{ll}1 & 0 \ 0 & 1end{array}right] ) and ( J= )
( left[begin{array}{cc}mathbf{0} & mathbf{1} \ -mathbf{1} & mathbf{0}end{array}right] )
Inverse of ( a I+b J ) is ( c I+d J ) if and only
if
( boldsymbol{a c}-boldsymbol{b d} neq mathbf{0} ) and ( boldsymbol{a} boldsymbol{d}+boldsymbol{b} boldsymbol{c}=mathbf{0} )
Reason
( (boldsymbol{a} boldsymbol{I}+boldsymbol{b} boldsymbol{J})(boldsymbol{c} boldsymbol{I}+boldsymbol{d} boldsymbol{J})=(boldsymbol{a} boldsymbol{c}-boldsymbol{b} boldsymbol{d}) boldsymbol{I}+ )
( (a d+b c) J )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
560 ( fleft(begin{array}{lll}1 & 5 & 2 \ 1 & -3 & 4end{array}right), ) then find ( A^{T} ) and ( left(A^{T}right)^{T} ) 12
561 Identify the incorrect statement in
respect of two square matrices ( A ) and ( B ) conformable for sum and product
A ( cdot t_{r}(A+B)=t_{r}(A)+t_{r}(B) )
B . ( t_{r}(alpha A)=alpha t_{r}(A), quad alpha epsilon R )
( mathbf{c} cdot t_{r}left(A^{T}right)=t_{r}(A) )
D. none of these
12
562 A and B are two square matrices of
same order and ( A^{prime} ) denotes the
transpose of ( A ), then
( mathbf{A} cdot(A B)^{prime}=B^{prime} A^{prime} )
B. ( (A B)^{prime}=A^{prime} B^{prime} )
( mathbf{C} cdot A B=0 Rightarrow|A|=0 ) or ( |B|=0 )
D . ( A B=0 Rightarrow A=0 ) or ( B=0 )
12
563 Let three matrices ( boldsymbol{A}=left[begin{array}{ll}mathbf{2} & mathbf{1} \ mathbf{4} & mathbf{1}end{array}right] ; boldsymbol{B}= )
( left[begin{array}{ll}mathbf{3} & mathbf{4} \ mathbf{2} & mathbf{3}end{array}right] ) and ( boldsymbol{C}=left[begin{array}{cc}mathbf{3} & -mathbf{4} \ -mathbf{2} & mathbf{3}end{array}right] ) then find
( operatorname{tr}(A)+operatorname{tr}left(frac{A B C}{2}right) operatorname{tr}left(frac{A(B C)^{2}}{4}right)+ )
( operatorname{tr}left(frac{A(B C)^{3}}{8}right)+ldots+infty, ) where ( t r(A) )
represents trace of matrix ( boldsymbol{A} )
( mathbf{A} cdot mathbf{6} )
B.
c. 12
D. 15
12
564 [
mathbf{A}=left[begin{array}{lll}
1 & 2 & 3 \
4 & 5 & 6 \
7 & 8 & 9
end{array}right]
]
The new matrix formed after
interchanging ( 2^{n d} ) and ( 3^{r d} ) rows will be
( A )
[
-left[begin{array}{lll}
1 & 2 & 3 \
4 & 5 & 6 \
7 & 8 & 9
end{array}right]
]
в.
[
left[begin{array}{lll}
4 & 5 & 6 \
1 & 2 & 3 \
7 & 8 & 9
end{array}right]
]
c.
[
-left[begin{array}{lll}
1 & 2 & 3 \
7 & 8 & 9 \
4 & 5 & 6
end{array}right]
]
D.
[
left[begin{array}{lll}
1 & 2 & 3 \
7 & 8 & 9 \
4 & 5 & 6
end{array}right]
]
12
565 If the matrices ( A, B,(A+B) ) are non singular then ( left[boldsymbol{A}(boldsymbol{A}+boldsymbol{B})^{-1} boldsymbol{B}right]^{-1} ) is
equal to-
( mathbf{A} cdot A+B )
B. ( A^{-1}+B^{-1} )
c. ( A(A+B)^{-1} )
D. None
12
566 If ( boldsymbol{A}=left[begin{array}{lll}mathbf{0} & mathbf{2} & mathbf{3} \ mathbf{3} & mathbf{5} & mathbf{7}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{lll}mathbf{1} & mathbf{3} & mathbf{7} \ mathbf{2} & mathbf{4} & mathbf{1}end{array}right] )
if ( boldsymbol{A}+boldsymbol{B}=left[begin{array}{ccc}mathbf{1} & mathbf{5} & mathbf{1 0} \ mathbf{5} & boldsymbol{k} & mathbf{8}end{array}right] )
Find the value of ( mathbf{k} )
( A cdot 9 )
B. 4
( c cdot 5 )
( D )
12
567 ( boldsymbol{B}=boldsymbol{A}+boldsymbol{A}^{2}+boldsymbol{A}^{3}+boldsymbol{A}^{4} )
If order of ( A ) is 3 then order of ( B ) is
( A cdot 3 )
B. 6
( c cdot 2 )
D. 9
12
568 ( mathbf{f} boldsymbol{A}=left[begin{array}{cc}mathbf{0} & mathbf{3} \ mathbf{2} & -mathbf{5}end{array}right] )
( & boldsymbol{K} boldsymbol{A}=left[begin{array}{cc}mathbf{0} & mathbf{4} boldsymbol{a} \ -mathbf{8} & mathbf{5} boldsymbol{b}end{array}right] )
then find the value of a and
12
569 Find the inverse of the matrix
( mathbf{A}=left[begin{array}{ccc}mathbf{0} & mathbf{1} & mathbf{2} \ {[mathbf{0 . 3 e m}] mathbf{1}} & mathbf{2} & mathbf{3} \ {[mathbf{0 . 3 e m}] mathbf{3}} & mathbf{1} & mathbf{1}end{array}right] )
12
570 If ( A B=A ) and ( B A=B ) then ( B^{2} ) is
equal to
A. ( B )
в. ( A )
( c .-B )
D. ( B^{2} )
12
571 Which of the following is not true, if ( mathbf{A} )
and ( B ) are two matrices each of order
( boldsymbol{n} times boldsymbol{n}, ) then
( mathbf{A} cdot(A+B)^{T}=B^{T}+A^{T} )
( mathbf{B} cdot(A-B)^{T}=A^{T}-B^{T} )
( mathbf{C} cdot(A B)^{T}=A^{T} B^{T} )
( mathbf{D} cdot(A B C)^{T}=C^{T} B^{T} A^{T} )
12
572 The inverse of a diagonal matrix is a :
This question has multiple correct options
A. Symmetric matrix
B. Skew-symmetric matrix
c. Diagonal matrix
D. None of the above
12
573 ( fleft(begin{array}{cc}cos alpha & sin alpha \ -sin alpha & cos alphaend{array}right], ) then find ( A^{2} ) 12
574 Unit matrix is a diagonal matrix in which all the diagonal elements are unity. Unit matrix of order ‘n’ is denoted
by ( I_{n}(text { or } I) ) i.e. ( A=left[a_{i j}right]_{n} ) is a unit
matrix when ( boldsymbol{a}_{boldsymbol{i} j}=mathbf{0} ) for ( boldsymbol{i} neq )
( boldsymbol{j} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{a}_{i j}=mathbf{1} )
A . True
B. False
12
575 The scalar matrixis
( mathbf{A} cdotleft[begin{array}{cc}-1 & 3 \ 2 & 4end{array}right] )
( mathbf{B} cdotleft[begin{array}{ll}0 & 3 \ 2 & 0end{array}right] )
( mathbf{C} cdotleft[begin{array}{ll}4 & 0 \ 0 & 4end{array}right] )
D. None of these
12
576 If ( A ) is symmetric and ( B ) is skew
symmetric matrix, then which of the following is/are correct? This question has multiple correct options
A ( cdot A B A^{T} ) is skew symmetric matrix
B. ( A B^{T}+B A^{T} ) is symmetric matrix
c. ( (A+B)(A-B) ) is symmetric
D. ( (A+I)(B-I) ) is skew symmetric
12
577 The inverse of a symmetric matrix is
A. Symmetric
B. Skew-symmetric
c. Diagonal
D. None of these
12
578 ( mathbf{f} boldsymbol{A}=left[begin{array}{ccc}mathbf{0} & boldsymbol{a}+mathbf{1} & boldsymbol{b}-mathbf{2} \ mathbf{2} boldsymbol{a}-mathbf{1} & mathbf{0} & boldsymbol{c}-mathbf{2} \ mathbf{2} boldsymbol{b}+mathbf{1} & mathbf{2}+boldsymbol{c} & mathbf{0}end{array}right] ) is skew
symmetric then ( a+b+c= )
( A cdot 3 )
B. -3
( c cdot frac{1}{3} )
( D )
12
579 Let ( boldsymbol{A}=left[begin{array}{c}mathbf{3} boldsymbol{x}^{2} \ mathbf{1} \ mathbf{6} boldsymbol{x}end{array}right], boldsymbol{B}=[boldsymbol{a}, boldsymbol{b}, boldsymbol{c}] ) and ( boldsymbol{C}= )
( left[begin{array}{ccc}(x+2)^{2} & 5 x^{2} & 2 x \ 5 x^{2} & 2 x & (x+2)^{2} \ 2 x & (x+2)^{2} & 5 x^{2}end{array}right] )
three given matrices, where ( a, b, c ) and
( boldsymbol{x} in boldsymbol{R}, ) Given that ( boldsymbol{t r} .(boldsymbol{A B})=boldsymbol{t r} .(boldsymbol{C}) vee )
( boldsymbol{x} in boldsymbol{R}, ) where ( boldsymbol{t r} .(boldsymbol{A}) ) denotes trace of ( boldsymbol{A} )
Find the value of ( (boldsymbol{a}+boldsymbol{b}+boldsymbol{c}) )
( A cdot 6 )
B.
( c cdot 8 )
D.
12
580 If ( A ) is a square matrix of order ( 3, ) then ( left|A d jleft(A d j A^{2}right)right|= )
A ( cdotleft|A^{2}right| )
B cdot ( left|A^{4}right| )
c. ( left|A^{8}right| )
( mathbf{D} cdotleft|A^{16}right| )
12
581 For any square matrix ( boldsymbol{A} ) with real numbers,
( A+A^{prime} ) is a symmetric and
( A-A^{prime} ) is a skew-symmetric
12
582 16. Let
lo 07 [
1007 To 107
0 0
P = I=/0 1 0 P = 0 0 1, P₂ = 1
Lo 0 1 0 1 1 0 0 1
To 107 To o 17 T0 0 17
Pa = 0 0 1 , Pg = 1 0 0 P6= 0 1 0
1 0 0 0 1 0 1 0 0
6 [2 1 37.
and X = P 1 0 2 PTS
k=1 [3 2 1
Where P denotes the transpose of the matrix P… Then
which of the following options is/are correct?
(a) X is a symmetric matrix
(JEE Adv. 2019)
(b) The sum of diagonal entries of X is 18
(C) X-301 is an invertible matrix
If X 1 = a 1, then a = 30
12
583 sin -1-sin?
23. Let M= 1+ cose cose
= a1 +BM-1
Where a= a(O) and B = B(O) are real numbers, and I is the
2 x 2 identity matrix. If a* is the minimum of the set
Sale): 0 E 10, 2TT)} and B* is the minimum of the set
SB(A):0 € 0, 21)}. Then the value of a*+b* is
(JEE Adv. 2019)
17
37
(a)
(b)
16
(©)
1
(d)
16
16
12
584 (10 o
26. Let A= 2 1 0 . Ifu, and u, are column matrices such
(3 2 1
12
585 ( left[begin{array}{cc}1 & 1 \ 0 & 1end{array}right]left[begin{array}{cc}1 & 2 \ 0 & 1end{array}right]left[begin{array}{cc}1 & 3 \ 0 & 1end{array}right] ldots ldotsleft[begin{array}{cc}1 & n \ 0 & 1end{array}right]= )
( A cdot 27 )
3.26
( c .376 )
D. 378
12
586 Find the inverse of the following matrix using elementary row transformation. ( left[begin{array}{cc}mathbf{1} & mathbf{2} \ mathbf{2} & -mathbf{1}end{array}right] ) 12
587 ( operatorname{Let} A=left[begin{array}{lll}1 & 0 & 0 \ 2 & 1 & 0 \ 3 & 2 & 1end{array}right] . ) If ( u_{1} ) and ( u_{2} ) are
column matrices such that ( boldsymbol{A} boldsymbol{u}_{1}=left[begin{array}{l}mathbf{1} \ mathbf{0} \ mathbf{0}end{array}right] )
and ( A u_{2}=left[begin{array}{l}0 \ 1 \ 0end{array}right] ) then ( u_{1}+u_{2} ) is equal to
( mathbf{A} cdotleft[begin{array}{c}-1 \ 1 \ 0end{array}right. )
B. ( left[begin{array}{c}1 \ -1 \ -1end{array}right. )
( mathbf{c} cdotleft[begin{array}{c}-1 \ -1 \ 0end{array}right. )
D. ( left[begin{array}{c}-1 \ 1 \ -1end{array}right. )
12
588 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{1} & mathbf{1}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{cc}mathbf{0} & -mathbf{1} \ mathbf{1} & mathbf{2}end{array}right], ) then
what is ( B^{-1} A^{-1} ) equal to?
begin{tabular}{ll}
A. ( left[begin{array}{cc}1 & -3 \
-3 & 2end{array}right] ) \
hline
end{tabular}
B. ( left[begin{array}{cc}-1 & 3 \ 1 & -2end{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}-1 & 3 \ -1 & -2end{array}right] )
D. ( left[begin{array}{cc}-1 & 1 \ -3 & -2end{array}right] )
12
589 If ( A ) is a ( 3 times 3 ) skew symmetric matrix,
then trace of ( A ) is equal to
A . -1
B. 7
c. ( |A| )
D. None of these
12
590 If ( P ) is a ( 3 times 3 ) matrix such that ( P^{T}= )
( 2 P+I, ) where ( P^{T^{prime}} ) is the transpose of ( P )
and ( I ) is the ( 3 times 3 ) identity matrix, then
there exists a column matrix ( boldsymbol{X}= )
( left[begin{array}{l}boldsymbol{x} \ boldsymbol{y} \ boldsymbol{z}end{array}right] neqleft[begin{array}{l}mathbf{0} \ mathbf{0} \ mathbf{0}end{array}right] ) such that
( ^{mathbf{A}} cdot operatorname{PX}=left[begin{array}{l}0 \ 0 \ 0end{array}right] )
в. ( P X=X )
c. ( P X=2 X )
D. ( P X=-X )
E.
[
P X=left[begin{array}{l}2 \ 0 \ 0end{array}right]
]
F. ( P X=3 X )
G. ( P X=5 X )
H. ( P X=-3 X )
12
591 ( boldsymbol{A}=left[begin{array}{cc}boldsymbol{x} & -mathbf{7} \ mathbf{7} & boldsymbol{y}end{array}right] ) is a skew-symmetric
matrix,
then ( (x, y)= )
A ( .(1,-1) )
B. (7,-7)
( c cdot(0,0) )
D. (14,-14)
12
592 A matrix consisting of a single column of m elements is know as
A. Column matrix
B. Row matrix
c. Square matrix
D. Null matrix
12
593 Let o be a complex cube root of unity with o # 1 and P=[P]
be a nx n matrix with p;;= mitj. Then p2 +0, when n=
(JEE Adv. 2013)
(a) 57 (6) 55 (c) 58 (d) 56
12
594 If ( A ) is a ( 3- ) rowed square matrix and
( |mathbf{3} boldsymbol{A}|=boldsymbol{k}|boldsymbol{A}| ) then ( boldsymbol{k}=? )
( A cdot 3 )
B. 9
( c cdot 27 )
D.
12
595 The table shows a five-day forecast indicating high (H) and Low(L) temperatures in Fahrenheit. Organise the temperatures in a matrix where the
first and second rows represent the High and Low temperatures respectively and identify which day will be the
warmest?
begin{tabular}{|c|c|c|c|c|}
hline Mon & Tue & wed & Thu & Fri \
hline( sum_{i}^{m} ) & ( sum_{i}^{m} ) & ( sum_{i}^{m} ) & ( sum_{i}^{m} ) & ( sum_{j}^{m} ) \
hline ( mathrm{H} 88 ) & ( mathrm{H} 90 ) & ( mathrm{H} 86 ) & ( mathrm{H} 84 ) & ( mathrm{H} 85 ) \
hline ( mathrm{L} 54 ) & ( mathrm{L} 56 ) & ( mathrm{L} 53 ) & ( mathrm{L} 52 ) & ( mathrm{L} 52 ) \
hline
end{tabular}
12
596 Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] )
whose element ( a_{i j} ) is ( a_{i j}=frac{(2 i+j)^{2}}{2} )
12
597 If ( boldsymbol{A}=left[begin{array}{ll}2 & 3 \ 5 & 7end{array}right], ) then find ( A+A^{T} ) 12
598 ff ( boldsymbol{A}=left[begin{array}{cc}cos boldsymbol{x} & sin boldsymbol{x} \ -sin boldsymbol{x} & cos boldsymbol{x}end{array}right], ) then find ( boldsymbol{x} )
satisfying ( mathbf{0}<boldsymbol{x}<frac{boldsymbol{pi}}{mathbf{2}} ) when ( boldsymbol{A}+boldsymbol{A}^{boldsymbol{T}}=boldsymbol{I} )
12
599 18.
Let 0 +1 be a cube root of unity and S be the set of all
[i a b
non-singular matrices of the form @ 1 c
02 @ 1
where each of a, b and c is either o or 02. Then the number
of distinct matrices in the set Sis
(2011)
(a) 2 (6) 6 (c) 4
(d) 8
12
600 If ( A ) is a ( 3 x 3 ) non singular matrix and
( |boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A}|=|boldsymbol{A}|^{x},|boldsymbol{a} boldsymbol{d} boldsymbol{j}(boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A})|= )
( |boldsymbol{A}|^{y},left|boldsymbol{A}^{-1}right|=|boldsymbol{A}|^{z} ) then the values of
( mathbf{x}, mathbf{y}, mathbf{z}, ) in descending order
A. ( mathrm{x}, mathrm{Y}, mathrm{z} )
в. Z, Y, х
c. ( mathrm{z}, mathrm{x}, mathrm{y} )
D. Y, x, z
12
601 f ( boldsymbol{y}=left[begin{array}{cc}1 & 2 \ -1 & 5end{array}right], ) find a matrix ( X ) such
that ( 2 boldsymbol{X}+boldsymbol{Y}=left[begin{array}{cc}mathbf{5} & mathbf{0} \ -mathbf{3} & mathbf{3}end{array}right] )
12
602 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{0} & mathbf{1} \ mathbf{1} & mathbf{0}end{array}right], ) then ( boldsymbol{A}^{2} ) is equal to
( A cdotleft[begin{array}{ll}0 & 1 \ 1 & 0end{array}right] )
B. ( left[begin{array}{ll}1 & 0 \ 1 & 0end{array}right] )
c. ( left[begin{array}{ll}1 & 0 \ 0 & 1end{array}right] )
D. ( left[begin{array}{ll}0 & 1 \ 0 & 1end{array}right] )
12
603 If ( A_{3} x_{3} ) and ( d e t A=2 ) then ( d e t A^{-1}= )
A ( cdot frac{1}{2} )
B. –
( c cdot frac{1}{4} )
D. –
12
604 If ( mathbf{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{0} & mathbf{3}end{array}right] ) and ( mathbf{B}=[mathbf{3}-mathbf{1}], ) then
( mathbf{B A}= )
A. ( left[begin{array}{ll}3 & 0 \ 0 & 3end{array}right] )
B. ( left[begin{array}{ll}3 & 0end{array}right] )
( begin{array}{ll}text { c. }[3 & 3]end{array} )
D. ( [0-3] )
12
605 If ( boldsymbol{A}=left[begin{array}{ccc}1 & -2 & 4 \ 2 & 3 & 2 \ 3 & 1 & 5end{array}right] ) and ( B= )
( left[begin{array}{ccc}mathbf{0} & -mathbf{2} & mathbf{4} \ mathbf{1} & mathbf{3} & mathbf{2} \ mathbf{- 1} & mathbf{1} & mathbf{5}end{array}right], ) then ( boldsymbol{A}+boldsymbol{B} ) is
( ^{A} cdotleft[begin{array}{ccc}1 & -2 & 4 \ 3 & 3 & 2 \ 2 & 1 & 5end{array}right] )
B. ( left[begin{array}{lll}1 & -2 & 8 \ 3 & 3 & 4 \ 2 & 1 & 10end{array}right] )
C ( cdotleft[begin{array}{lll}1 & -4 & 8 \ 3 & 6 & 4 \ 2 & 2 & 10end{array}right] )
D. none of these
12
606 If ( boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi}, ) then
( left|begin{array}{ccc}sin (boldsymbol{A}+boldsymbol{B}+boldsymbol{C}) & sin boldsymbol{B} & cos boldsymbol{C} \ sin boldsymbol{theta} & tan boldsymbol{A} \ cos (boldsymbol{A}+boldsymbol{B}) & -tan boldsymbol{A} & boldsymbol{0}end{array}right|= )
12
607 The order of ( [mathbf{x} mathbf{y} mathbf{z}]left[begin{array}{lll}mathbf{a} & mathbf{h} & mathbf{g} \ mathbf{h} & mathbf{b} & mathbf{f} \ mathbf{g} & mathbf{f} & mathbf{c}end{array}right]left[begin{array}{l}mathbf{x} \ mathbf{y} \ mathbf{z}end{array}right] )
is
A . ( 3 times 1 )
B. 1x
( c .1 times 3 )
D. 3×3
12
608 ( mathbf{f} boldsymbol{A}=left[begin{array}{lll}mathbf{1} & mathbf{4} & mathbf{0} \ mathbf{2} & mathbf{5} & mathbf{0} \ mathbf{3} & mathbf{6} & mathbf{0}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{lll}mathbf{3} & mathbf{2} & mathbf{1} \ mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{4} & mathbf{5} & mathbf{6}end{array}right] )
and ( C=left[begin{array}{lll}3 & 2 & 1 \ 1 & 2 & 3 \ 7 & 8 & 9end{array}right], ) Then evaluate
( boldsymbol{A B}-boldsymbol{B C} )
12
609 Which of the following matrix is inverse of itself
A. ( left[begin{array}{lll}1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1end{array}right] )
В. ( left[begin{array}{lll}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1end{array}right] )
C. ( left[begin{array}{lll}1 & 0 & 1 \ 0 & 0 & 0 \ 1 & 0 & 1end{array}right] )
D. ( left[begin{array}{lll}0 & 1 & 0 \ 1 & 1 & 1 \ 0 & 1 & 0end{array}right] )
12
610 If ( A ) is a skew-symmetric matrix of
order ( 3, ) then prove that det ( A=0 )
12
611 Show that all the diagonal elements of a skew-symmetric matrix are zero. 12
612 If ( 2left[begin{array}{ll}1 & 3 \ 0 & xend{array}right]+left[begin{array}{ll}y & 0 \ 1 & 2end{array}right]=left[begin{array}{ll}5 & 6 \ 1 & 8end{array}right], ) then
the value of ( x ) and ( y ) are
A ( . x=3, y=3 )
в. ( x=-3, y=3 )
c. ( x=3, y=-3 )
D. ( x=-3, y=-3 )
12
613 Find ( boldsymbol{x}, boldsymbol{y} ) if ( left[begin{array}{cc}mathbf{0} & mathbf{4} \ boldsymbol{x}^{2} & boldsymbol{y}^{2}end{array}right]=left[begin{array}{ll}mathbf{0} & mathbf{4} \ boldsymbol{4} & mathbf{9}end{array}right] ) 12
614 ff ( P(x)=left[begin{array}{cc}cos x & sin x \ -sin x & cos xend{array}right] ) then
( boldsymbol{P}(boldsymbol{x}) cdot boldsymbol{P}(boldsymbol{y})= )
This question has multiple correct options
A ( . P(x) . P(y)=P(x+y) )
в. ( P(x) . P(y)=P(x y) )
c. ( P(x) . P(y)=P(y) . P(x) )
12
615 For the matrices ( A ) and ( B ), verify that
( (A B)^{prime}=B^{prime} A^{prime} ) where
(i) ( boldsymbol{A}=left[begin{array}{c}mathbf{1} \ -mathbf{4} \ mathbf{3}end{array}right], boldsymbol{B}=left[begin{array}{lll}-mathbf{1} & mathbf{2} & mathbf{1}end{array}right] )
(ii) ( boldsymbol{A}=left[begin{array}{l}mathbf{0} \ mathbf{1} \ mathbf{2}end{array}right], boldsymbol{B}=left[begin{array}{lll}mathbf{1} & mathbf{5} & mathbf{7}end{array}right] )
12
616 If ( boldsymbol{P}=left[begin{array}{cc}frac{sqrt{mathbf{3}}}{mathbf{2}} & frac{mathbf{1}}{mathbf{2}} \ -frac{mathbf{1}}{mathbf{2}} & frac{sqrt{mathbf{3}}}{mathbf{2}}end{array}right], boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{1} \ mathbf{0} & mathbf{1}end{array}right] ) and
( boldsymbol{Q}=boldsymbol{P} boldsymbol{A} boldsymbol{P}^{T}, ) then ( boldsymbol{P}^{T} boldsymbol{Q}^{2015} boldsymbol{P} ) is:
A. ( left[begin{array}{cc}0 & 2015 \ 0 & 0end{array}right] )
B. ( left[begin{array}{cc}2015 & 0 \ 1 & 2015end{array}right] )
c. ( left[begin{array}{cc}2015 & 1 \ 0 & 2015end{array}right] )
D. ( left[begin{array}{cc}1 & 2015 \ 0 & 1end{array}right] )
12
617 fthe matrix ( left[begin{array}{cc}mathbf{2} & mathbf{3} \ mathbf{5} & -mathbf{1}end{array}right]=boldsymbol{A}+boldsymbol{B}, ) where
A is symmetric and B is skew symmetric, then ( B= )
A. ( left[begin{array}{cc}2 & 4 \ 4 & -1end{array}right] )
B. ( left[begin{array}{cc}0 & -2 \ 2 & 0end{array}right] )
C ( cdotleft[begin{array}{cc}0 & 1 \ -1 & 0end{array}right] )
D. ( left[begin{array}{cc}0 & -1 \ 1 & 0end{array}right] )
12
618 Simplify ( : cos Qleft[begin{array}{cc}cos Q & sin Q \ -sin Q & cos Qend{array}right]+ )
( sin Qleft[begin{array}{cc}sin Q & -cos Q \ cos Q & sin Qend{array}right] )
12
619 Given the matrices
[
begin{array}{l}
boldsymbol{A}=left[begin{array}{lll}
2 & 1 & 1 \
3 & -1 & 0 \
0 & 2 & 4
end{array}right], B=left[begin{array}{lll}
9 & 7 & -1 \
3 & 5 & 4 \
2 & 1 & 6
end{array}right] \
text { and } boldsymbol{C}=left[begin{array}{lll}
2 & -4 & 3 \
1 & -1 & 0 \
9 & 4 & 5
end{array}right]
end{array}
]
Verify that ( (boldsymbol{A}+boldsymbol{B})+boldsymbol{C}=boldsymbol{A}+(boldsymbol{B}+ )
( C) )
12
620 If ( 2 A-left[begin{array}{ll}1 & 2 \ 7 & 4end{array}right]=left[begin{array}{cc}3 & 0 \ 0 & -2end{array}right], ) then ( A ) is
equal to
( mathbf{A} cdotleft[begin{array}{cc}2 & 1 \ 7 / 2 & 1end{array}right] )
В. ( left[begin{array}{cc}4 & 4 \ 7 / 2 & 1end{array}right] )
c. ( left[begin{array}{ll}3 & -1 \ 7 & 2end{array}right] )
D. None of these
12
621 If ( boldsymbol{A}=left[begin{array}{cc}cos boldsymbol{x} & sin boldsymbol{x} \ -sin boldsymbol{x} & cos boldsymbol{x}end{array}right] ) and
( boldsymbol{A}(boldsymbol{A} boldsymbol{d} boldsymbol{j} boldsymbol{A})=boldsymbol{k}left[begin{array}{ll}mathbf{1} & mathbf{0} \ mathbf{0} & mathbf{1}end{array}right] ) then the value of
( k ) is
( mathbf{A} cdot sin x cos x )
B.
c.
D.
12
622 Find the inverse of the following matrix
using transformation method. ( left[begin{array}{cc}2 & -3 \ -1 & 2end{array}right] )
12
623 By using elementary transformation find the inverce of ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{2}-mathbf{1}end{array}right] ) 12
624 If the matrix ( left[begin{array}{ccc}mathbf{0} & mathbf{2} boldsymbol{beta} & mathbf{Upsilon} \ boldsymbol{alpha} & boldsymbol{beta} & -mathbf{Upsilon} \ boldsymbol{alpha} & -boldsymbol{beta} & mathbf{Upsilon}end{array}right] ) is
orthogonal, then
A ( cdot alpha=pm frac{1}{sqrt{2}} )
в. ( beta=pm frac{1}{sqrt{6}} )
c. ( _{gamma}=pm frac{1}{sqrt{3}} )
D. all of these
12
625 If the system of equations ( 3 x-2 y+ )
( z=0, lambda x-14 y+15 z=0, x+2 y- )
( mathbf{3} z=mathbf{0} ) have non zero solution, then find
( lambda )
12
626 ff ( mathbf{A}=left[begin{array}{cccc}cos & boldsymbol{theta} & sin & boldsymbol{theta} \ boldsymbol{s i n} & boldsymbol{theta} & -boldsymbol{operatorname { c o s }} & boldsymbol{theta}end{array}right], mathbf{B}=left[begin{array}{cc}mathbf{1} & mathbf{0} \ -mathbf{1} & mathbf{1}end{array}right] )
( mathbf{C}=boldsymbol{A} boldsymbol{B} boldsymbol{A}^{boldsymbol{T}} )
then ( A^{T} C A ) equals to ( (n quad in quad N) )
A. ( left[begin{array}{cc}-n & 1 \ 1 & 0end{array}right] )
B. ( left[begin{array}{cc}1 & -n \ 0 & 1end{array}right] )
C. ( left[begin{array}{cc}0 & 1 \ 1 & -nend{array}right] )
D. ( left[begin{array}{cc}1 & 0 \ -n & 1end{array}right] )
12
627 Let ( boldsymbol{x} in boldsymbol{R} ) and let
( boldsymbol{P}=left[begin{array}{lll}1 & 1 & 1 \ 0 & 2 & 2 \ 0 & 0 & 3end{array}right], Q=left[begin{array}{lll}2 & x & x \ 0 & 4 & 0 \ x & x & 6end{array}right] ) and
( boldsymbol{R}=boldsymbol{P Q P}^{-1} )
Then which of the following is are
correct

This question has multiple correct options
A. there exists a real number x such that ( P Q=Q P )
B.
det ( R=operatorname{det}left[begin{array}{lll}2 & x & x \ 0 & 4 & 0 \ x & x & 5end{array}right]+8 ) for all ( x in R )
C. For ( x=1 ) there exists a unit vector ( alpha hat{i}+beta hat{j}+gamma hat{k} ) for
which are ( Rleft[begin{array}{l}alpha \ beta \ gammaend{array}right]=left[begin{array}{l}0 \ 0 \ 0end{array}right] )
D. ( quad ) For ( x=0 ) if ( Rleft[begin{array}{l}1 \ a \ bend{array}right]=6left[begin{array}{l}1 \ a \ bend{array}right] ) then ( a+b= )

12
628 ( boldsymbol{A}=left[begin{array}{cc}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{2} \ mathbf{- 1} & mathbf{0}end{array}right], boldsymbol{B}=left[begin{array}{ccc}mathbf{1} & mathbf{3} & mathbf{2} \ mathbf{4} & mathbf{- 1} & mathbf{3}end{array}right], ) then
order of ( A B ) is
A. ( 2 times 2 )
в. ( 3 times 3 )
( c cdot 1 times 3 )
D. ( 3 times 2 )
12
629 Find the inverse of the following matrices by the adjoining method ( left[begin{array}{ccc}1 & 0 & 0 \ 3 & 3 & 0 \ 5 & 2 & -1end{array}right] ) 12
630 ( mathbf{f} mathbf{P}=left{begin{array}{cc}frac{sqrt{3}}{2} & frac{1}{2} \ -frac{1}{2} & frac{sqrt{3}}{2}end{array}right} mathbf{A}=left{begin{array}{ll}mathbf{1} & mathbf{1} \ mathbf{0} & mathbf{1}end{array}right} )
and ( mathbf{Q}=mathbf{P} mathbf{A} mathbf{P}^{mathbf{T}} ) and ( mathbf{x}=mathbf{P}^{mathbf{T}} mathbf{Q}^{2005} mathbf{P} )
then ( x ) is equal to
A. ( left{begin{array}{ll}1 & 2005 \ 0 & 1end{array}right} )
в. ( left{begin{array}{ll}4+2005 sqrt{3} & 6015 \ 2005 & 4-2005 sqrt{3}end{array}right} )
c. ( _{frac{1}{4}}left[begin{array}{cc}2+sqrt{3} & 1 \ -1 & 2-sqrt{3}end{array}right] )
D. ( frac{1}{4}left{begin{array}{ll}2005 & 2-sqrt{3} \ 2+sqrt{3} & 2005end{array}right} )
12
631 If ( boldsymbol{A}=left(begin{array}{lll}1 & 2 & 2 \ 2 & 1 & 2 \ 2 & 2 & 1end{array}right) )
If ( A^{2}-4 A=p I ) where ( I ) and ( O ) are the
unit matrix and the null matrix of order
3 respectively. Find the value of ( p )
A. ( p=2 )
в. ( p=3 )
( mathbf{c} cdot p=4 )
D. ( p=5 )
12
632 If ( boldsymbol{A}=left[begin{array}{cc}2 & boldsymbol{4} \ -mathbf{1} & boldsymbol{k}end{array}right] ) and ( boldsymbol{A}^{2}=mathbf{0}, ) find ( boldsymbol{k} ) 12
633 If ( boldsymbol{A}=left[begin{array}{cr}mathbf{2} & mathbf{3} \ -mathbf{1} & mathbf{4}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{cc}-mathbf{3} & mathbf{1} \ mathbf{4} & -mathbf{2}end{array}right] )
then find ( boldsymbol{A}-boldsymbol{B} )
12
634 Q Type your question
Model 18 el ( x quad 20 )
Model 5 el
Model ( z ) 19 11
The table above shows the number of TV
sets that were sold during a three-day
sale. The prices of models ( X, Y ) and ( Z )
( operatorname{are} $ 99, $ 199, ) and ( $ 299, ) respectively.
Which of the following matrix
representations gives the total income,
in dollars, received from the sale of the
TV sets for each of the three days?
( mathbf{A} )
( begin{array}{ll}199 & 299end{array} ) ( left[begin{array}{ccc}20 & 18 & 3 \ 16 & 5 & 8 \ 19 & 11 & 10end{array}right][99 )
B. ( left[begin{array}{ccc}20 & 18 & 3 \ 16 & 5 & 8 \ 19 & 11 & 10end{array}right]left[begin{array}{c}99 \ 199 \ 299end{array}right] )
( c )
( 199 quad 2 )
99
299]( left[begin{array}{ccc}20 & 18 & 3 \ 16 & 5 & 8 \ 19 & 11 & 10end{array}right] )
D. ( left[begin{array}{c}99 \ 199 \ 299end{array}right] quadleft[begin{array}{ccc}20 & 18 & 3 \ 16 & 5 & 8 \ 19 & 11 & 10end{array}right] )
E . ( bullet cdot 199left[begin{array}{ccc}20 & 18 & 3 \ 16 & 5 & 8 \ 19 & 11 & 10end{array}right]+299left[begin{array}{ccc}20 & 18 & 3 \ 16 & 5 & 8 \ 19 & 11 & 10end{array}right] )
12
635 Let ( A ) and ( B ) be two symmetic matrices
of order 3
Statement ( -1: boldsymbol{A}(boldsymbol{B} boldsymbol{A}) ) and ( (boldsymbol{A B}) boldsymbol{A} ) are
symmetric matrices.
Statement – ( 2: A B ) is symmetric matrix if matrix multiplication of ( boldsymbol{A} ) and ( boldsymbol{B} ) is
commutative.
A. Statement-1 is True, Statement-2 is True, Statementis a correct explanation for Statement-
B. Statement-1 is True, Statement-2 is True, Statementis NOT a correct explanation for Statement-
c. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True
12
636 If ( A ) and ( B ) are matrices of the same
order, then ( A B^{T}-B A^{T} ) is a
A. Skew-symmetric matrix
B. Null matrix
c. Unit matrix
D. symmetric matrix
12
637 Two ( n times n ) square matrices ( A ) and ( B ) are
said to be similar if there exists a non-
singular matrix ( boldsymbol{P} ) such that
( boldsymbol{P}^{-1} boldsymbol{A} boldsymbol{P}=boldsymbol{B} )
If ( A ) and ( B ) are similar matrices such
that ( operatorname{det}(A)=1, ) then
A. ( operatorname{det}(B)=1 )
B. ( operatorname{det}(A)+operatorname{det}(B)=0 )
c. ( operatorname{det}(B)=-1 )
D. none of these
12
638 Two ( n times n ) square matrices ( A ) and ( B ) are
said to be similar if there exists a non-
singular matrix ( boldsymbol{P} ) such that
( boldsymbol{P}^{-1} boldsymbol{A} boldsymbol{P}=boldsymbol{B} )
If ( A ) and ( B ) are similar and ( B ) and ( C ) are
similar, then
A. ( A B ) and ( B C ) are similar
B. ( A ) and ( C ) are similar
c. ( A+C ) and ( B ) are similar
D. none of these
12
639 If ( A ) be a skew symmetric matrix of
order ( m ) than ( A+A^{prime} ) is a
A. Nilpotent matrix
B. Orthogonal matrix
c. Null matrix
D. Skew symmetric
12
640 Matrices ( A ) and ( B ) will be inverse of
each other only if
( mathbf{A} cdot A B=B A )
в. ( A B=0, B A=I )
c. ( A B=B A=0 )
D. ( A B=B A=I )
12
641 What is the order of the product ( left[begin{array}{lll}boldsymbol{x} & boldsymbol{y} & boldsymbol{z}end{array}right]left[begin{array}{lll}boldsymbol{a} & boldsymbol{h} & boldsymbol{g} \ boldsymbol{h} & boldsymbol{b} & boldsymbol{f} \ boldsymbol{g} & boldsymbol{f} & boldsymbol{c}end{array}right]left[begin{array}{l}boldsymbol{x} \ boldsymbol{y} \ boldsymbol{z}end{array}right] )
( A cdot 3 times 1 )
в. ( 1 times 1 )
( c cdot 1 times 3 )
D. ( 3 times 3 )
12
642 If ( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{9} & mathbf{- 7} \ boldsymbol{i} & boldsymbol{omega}^{boldsymbol{n}} & mathbf{8} \ mathbf{1} & boldsymbol{6} & boldsymbol{omega}^{mathbf{2 n}}end{array}right] ) where ( boldsymbol{i}=sqrt{-mathbf{1}} )
and ( omega ) is complex cube root of unity, then ( operatorname{tr}(A) ) will be
This question has multiple correct options
A. ( 1, ) if ( n=3 k, k in N )
B. ( 3, ) if ( n=3 k, k in N )
c. ( 0, ) if ( n neq 3 k, k epsilon in N )
D. ( -1, ) if ( n neq 3 k, k epsilon in N )
12
643 Express the square matrix ( A ) as the
sum of a symmetric and a skewsymmetric matrix.
12
644 Find the values of ( x, y ) and ( z ) if ( left[begin{array}{lll}boldsymbol{x} & mathbf{5} & mathbf{4} \ mathbf{5} & mathbf{9} & mathbf{1}end{array}right]=left[begin{array}{lll}mathbf{3} & mathbf{5} & boldsymbol{z} \ mathbf{5} & boldsymbol{y} & mathbf{1}end{array}right] ) 12
645 Using elementary tansormations, find
the inverse of each of the matrices, if it
exists in ( left[begin{array}{ll}1 & 3 \ 2 & 7end{array}right] )
12
646 If ( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{- 1} & mathbf{2} \ mathbf{3} & mathbf{0} & -mathbf{2} \ mathbf{2} & mathbf{0} & mathbf{3}end{array}right] ) then ( (boldsymbol{a} boldsymbol{d} boldsymbol{j} boldsymbol{A}) boldsymbol{A}= )
( mathbf{A} cdotleft[begin{array}{ccc}13 & 0 & 0 \ 0 & 13 & 0 \ 0 & 0 & 13end{array}right] )
( mathbf{B} cdotleft[begin{array}{ccc}7 & 0 & 0 \ 0 & 7 & 0 \ 0 & 0 & 7end{array}right] )
( mathbf{C} cdotleft[begin{array}{ccc}-7 & 0 & 0 \ 0 & 0 & -7 \ 0 & 0 & -7end{array}right] )
D. ( left[begin{array}{ccc}11 & 0 & 0 \ 0 & 11 & 0 \ 0 & 0 & 11end{array}right] )
12
647 There are 6 Higher Secondary Schools, 8 High Schools and 13 Primary Schools in a town. Represent these data in the form
of ( 3 times 1 ) and ( 1 times 3 ) matrices.
12
648 f ( boldsymbol{A}=operatorname{diag}(mathbf{2}-mathbf{5} mathbf{9}), boldsymbol{B}=operatorname{diag}(mathbf{1} mathbf{1}- )
4) and ( C=operatorname{diag}(-634), ) then find ( B+ )
( C-2 A )
12
649 If ( mathbf{2} boldsymbol{A}+boldsymbol{B}=left[begin{array}{cc}mathbf{1} & mathbf{-} mathbf{1} \ mathbf{0} & mathbf{1} \ mathbf{1} & mathbf{-} mathbf{2}end{array}right] boldsymbol{A}- )
( mathbf{2} boldsymbol{B}=left[begin{array}{cc}mathbf{0} & mathbf{1} \ mathbf{- 2} & mathbf{0} \ mathbf{1} & mathbf{- 1}end{array}right] ) find ( boldsymbol{A} ) and ( boldsymbol{B} )
12
650 Find the inverse of the matrix
( $ $ mathrm{~ l l e f t [ l b e g i n ~ { a r r a y } ~} mid 5 & 11 backslash 14 & 9 )
( mathrm{~ l e n d ~ { a r r a y } l r i g h t ] $ $ ~ b y ~ e l e m e n t a r y ~} )
transformations.
12
651 The set of natural numbers is divided
into arrays of rows and columns in the
form of matrices as ( boldsymbol{A}_{1}=(1), boldsymbol{A}_{2}=left(begin{array}{ll}2 & 3 \ 4 & 5end{array}right), boldsymbol{A}_{3}= )
( left(begin{array}{ccc}mathbf{6} & mathbf{7} & mathbf{8} \ mathbf{9} & mathbf{1 0} & mathbf{1 1} \ mathbf{1 2} & mathbf{1 3} & mathbf{1 4}end{array}right) ldots mathbf{s o} ) on
Find the value of ( boldsymbol{T}_{boldsymbol{r}}left(boldsymbol{A}_{mathbf{1 0}}right) )
( left[mathrm{Note}: T_{r}(A) ) denotes sum of diagonal right.
elements of ( boldsymbol{A} . )
A . 3355
в. 3434
( c .5533 )
D. None of these
12
652 Find the inverse of the matrix
( left[begin{array}{ccc}1 & 0 & 0 \ 3 & 3 & 0 \ 5 & 2 & 1end{array}right] )
12
653 42. If [ 1 17.51 27 51 37
If ſo 1] [o 13o 1…
[1 n-1] [1 787
Lo 1 )=l6 78).
then the inverse of o 1 is: [JEE M 2019 –9 April (M]
1
-17
12
654 Let ( boldsymbol{A}=left[begin{array}{cc}mathbf{2} & mathbf{4} \ mathbf{3} & mathbf{2}end{array}right], boldsymbol{B}=left[begin{array}{cc}mathbf{1} & mathbf{3} \ -mathbf{2} & mathbf{5}end{array}right] ) and
( boldsymbol{C}=left[begin{array}{cc}mathbf{- 2} & mathbf{5} \ mathbf{3} & mathbf{4}end{array}right] . ) Find:
( boldsymbol{A}-mathbf{2} boldsymbol{B}+mathbf{3} boldsymbol{C} )
12
655 If ( A ) is ( 2 times 2 ) matrix such that ( A^{2}=0 )
then ( t r(A) ) is
A
B.
( c cdot-1 )
D. none of these
12
656 If the number of elements in a matrix is
60 then how many different order of matrix are possible
A . 12
B. 6
( c cdot 24 )
D. none of these
12
657 If ( A ) is ( 2 times 3 ) matrix and ( A B ) is a ( 2 times 5 )
matrix, then ( B ) must be a
A. ( 3 times 5 ) matrix
B. ( 5 times 3 ) matrix
c. ( 3 times 2 ) matrix
D. ( 5 times 2 ) matrix
12
658 If ( P=left[begin{array}{lll}4 & 3 & 2end{array}right] ) and ( Q=left[begin{array}{lll}-1 & 2 & 3end{array}right] ) then
( P-Q= )
( left.begin{array}{lll}text { A } cdot[6 & -1 & -4end{array}right] )
B . ( left[begin{array}{lll}2 & -1 & -4end{array}right] )
( mathbf{c} cdotleft[begin{array}{lll}6 & 1 & 4end{array}right] )
D・[-4 -1 6]
12
659 Consider three matrices ( A= ) ( left[begin{array}{ll}mathbf{2} & mathbf{1} \ mathbf{4} & mathbf{1}end{array}right] quad boldsymbol{B}=left[begin{array}{ll}mathbf{3} & mathbf{4} \ mathbf{2} & mathbf{3}end{array}right] quad ) and ( quad boldsymbol{C}= )
( left[begin{array}{cc}mathbf{3} & -mathbf{4} \ -mathbf{2} & mathbf{3}end{array}right] )
The value of the sum ( t r(A)+ ) ( operatorname{tr}left(frac{A B C}{2}right)+operatorname{tr}left(frac{A(B C)^{2}}{4}right)+ )
( operatorname{tr}left(frac{boldsymbol{A}(boldsymbol{B C})^{3}}{boldsymbol{8}}right)+ldots ldots ldots .+infty ) is
( (operatorname{tr}(A) text { denotes trace of a matrix } A) )
A . 6
B. 9
c. 12
D. none of these
12
660 If ( boldsymbol{A}=left(begin{array}{ll}mathbf{3} & mathbf{5} \ mathbf{7} & mathbf{9}end{array}right) ) is written as ( boldsymbol{A}=boldsymbol{P}+boldsymbol{Q} )
where ( boldsymbol{P} ) is a symmetric matrix and ( boldsymbol{Q} ) is skew-symmetric matrix, then write the ( operatorname{matrix} boldsymbol{P} )
12
661 If ( boldsymbol{A}=left[begin{array}{ccc}boldsymbol{4} & boldsymbol{1} & boldsymbol{0} \ boldsymbol{1} & boldsymbol{-} boldsymbol{2} & boldsymbol{2}end{array}right], boldsymbol{B}= )
( left[begin{array}{ccc}boldsymbol{2} & boldsymbol{0} & -boldsymbol{1} \ boldsymbol{3} & boldsymbol{1} & boldsymbol{4}end{array}right], boldsymbol{C}=left[begin{array}{c}boldsymbol{1} \ boldsymbol{2} \ -mathbf{1}end{array}right] ) and ( left(boldsymbol{3} boldsymbol{B}-boldsymbol{t}_{boldsymbol{}}right) )
( boldsymbol{2} boldsymbol{A}) boldsymbol{C}+boldsymbol{2} boldsymbol{X}=boldsymbol{0} operatorname{then} boldsymbol{X}= )
A ( cdot frac{1}{2}left[begin{array}{c}3 \ 13end{array}right] )
B ( cdot frac{1}{2}left[begin{array}{c}3 \ -13end{array}right] )
c. ( frac{1}{2}left[begin{array}{c}-3 \ 13end{array}right] )
D. ( left[begin{array}{c}3 \ -13end{array}right] )
12
662 The number of different possible orders of matrices having 18 identical elements is
( A cdot 3 )
B.
( c cdot 6 )
D. 4
12
663 If ( A ) is a square matrix with ( |A|=8 )
Find the value of ( left|boldsymbol{A} boldsymbol{A}^{-1}right| )
12
664 A square non-singular matrix A satisfies ( boldsymbol{A}^{2}-boldsymbol{A}+boldsymbol{2} boldsymbol{I}=boldsymbol{0}, ) then ( boldsymbol{A}^{-1}= )
( mathbf{A} cdot I-A )
B ( cdot frac{1}{2}(I-A) )
c. ( I+A )
D ( cdot frac{1}{2}(I+A) )
12
665 ( mathbf{f} A^{prime}=left[begin{array}{cc}mathbf{3} & mathbf{4} \ -mathbf{1} & mathbf{2} \ mathbf{0} & mathbf{1}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{ccc}-mathbf{1} & mathbf{2} & mathbf{1} \ mathbf{1} & mathbf{2} & mathbf{3}end{array}right] )
then verify that:
( (i)(A+B)^{prime}=A^{prime}+B^{prime} )
( (i i)(A-B)^{prime}=A^{prime}-B^{prime} )
12
666 If ( boldsymbol{A} ) is a skew symmetric matrix of even order then ( operatorname{det}(A) ) is
( mathbf{A} cdot mathbf{0} )
B ( . neq 0 )
c. non zero perfect square
D. None of these
12
667 The number of ( A ) in ( T_{p} ) such that the
trace of ( A ) is not divisible by ( p ) but
( operatorname{det}(A) ) is divisible by ( p ) is? [Note: The trace of a matrix is the sum of its diagonal entries.
A ( cdot(p-1)left(p^{2}-p+1right) )
В . ( p^{3}-(p-1)^{2} )
c. ( (p-1)^{2} )
D. ( (p-1)left(p^{2}-2right) )
12
668 [
text { If } boldsymbol{A}=left[begin{array}{cc}
mathbf{2} & mathbf{2} \
mathbf{- 3} & mathbf{1} \
mathbf{4} & mathbf{0}
end{array}right], boldsymbol{B}=left[begin{array}{cc}
mathbf{6} & mathbf{2} \
mathbf{1} & mathbf{3} \
mathbf{0} & mathbf{4}
end{array}right], text { find }
]
matrix ( C ) such that ( A+B+C=0 )
where ( boldsymbol{O} ) is the zero matrix.
12
669 If ( X ) is a ( 2 times 3 ) matrix such that
( left|boldsymbol{X}^{boldsymbol{T}} boldsymbol{X}right| neq mathbf{0} ) and ( boldsymbol{A}=boldsymbol{I}_{2}- )
( Xleft(X^{T} Xright)^{-1} X^{T} ) then ( A^{2} ) is equal to:
( left(X^{T} text { denotes transpose of matrix } Xright) )
A. ( A )
B. ( I )
( c cdot A^{-1} )
D. ( A X )
12
670 If ( A ) and ( B ) are square matrices of same
order, then which of the following is
correct –
A. ( A+B=B+A )
в. ( A+B=A B )
c. ( A B=B A )
D. ( A B=B+A )
12
671 ff ( D=left|begin{array}{ccc}a_{1} & b_{1} & c_{1} \ a_{2} & b_{2} & c_{2} \ a_{3} & b_{3} & c_{3}end{array}right| ) and ( D_{0}= )
( left|begin{array}{ccc}boldsymbol{k} boldsymbol{a}_{1} & boldsymbol{k} boldsymbol{b}_{1} & boldsymbol{k} boldsymbol{c}_{1} \ boldsymbol{k} boldsymbol{a}_{2} & boldsymbol{k} boldsymbol{b}_{2} & boldsymbol{k} boldsymbol{c}_{2} \ boldsymbol{k} boldsymbol{a}_{3} & boldsymbol{k} boldsymbol{b}_{3} & boldsymbol{k} boldsymbol{c}_{3}end{array}right| ) then show that
( boldsymbol{D}_{0}=boldsymbol{k}^{3} boldsymbol{D} )
12
672 Assertion ( operatorname{Let} boldsymbol{A}=left[begin{array}{ll}boldsymbol{a}_{11} & boldsymbol{a}_{12} \ boldsymbol{a}_{21} & boldsymbol{a}_{22}end{array}right], boldsymbol{X}=left[begin{array}{l}boldsymbol{x}_{1} \ boldsymbol{x}_{2}end{array}right], boldsymbol{Y}= )
( left[begin{array}{l}boldsymbol{y}_{1} \ boldsymbol{y}_{2}end{array}right] )
If ( boldsymbol{X}^{prime} boldsymbol{A} boldsymbol{X}=boldsymbol{0} ) for each ( boldsymbol{X}, ) then ( boldsymbol{A} ) must
be a symmetric matrix.
Reason
If ( boldsymbol{A} ) is symmetric and ( boldsymbol{X}^{prime} boldsymbol{A} boldsymbol{X}=mathbf{0} ) for
( operatorname{each} X, ) then ( A=0 )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
673 The matrix A satisfies the matrix
equation if ( A=left[begin{array}{lll}1 & 2 & 2 \ 2 & 1 & 2 \ 2 & 2 & 1end{array}right] )
( mathbf{A} cdot A^{2}-4 A-5 I=0 )
B ( cdot A^{2}-4 A-5=0 )
C ( cdot A^{2}+4 A-5 I=0 )
D ( cdot A^{2}+4 A-5=0 )
12
674 Construct a ( 3 times 2 ) matrix ( A=left[a_{i j}right] )
whose elements are given by ( boldsymbol{a}_{boldsymbol{i} j}= ) ( frac{(i-2 j)^{2}}{2} )
12
675 ( boldsymbol{A}=left(begin{array}{cc}mathbf{5} & mathbf{2} \ mathbf{7} & mathbf{3}end{array}right) ) and ( boldsymbol{B}=left(begin{array}{cc}mathbf{2} & mathbf{- 1} \ mathbf{- 1} & mathbf{2}end{array}right) )
verify that ( (boldsymbol{A B})^{boldsymbol{T}}=boldsymbol{B}^{boldsymbol{T}} boldsymbol{A}^{boldsymbol{T}} )
12
676 If ( boldsymbol{A}=left[begin{array}{cc}-mathbf{3} & mathbf{5} \ mathbf{5} & mathbf{0} \ -mathbf{7} & mathbf{4}end{array}right] ) and ( boldsymbol{B}= )
( left[begin{array}{ccc}mathbf{3} & mathbf{- 5} & mathbf{7} \ mathbf{- 5} & mathbf{0} & mathbf{- 4}end{array}right], ) then find ( boldsymbol{A}+boldsymbol{B}^{boldsymbol{T}} )
A . 0
в. ( 2 B )
c. ( 2 A^{text {न }} )
D. ( 2 B^{text {T }} )
12
677 Let ( k ) be a positive real number and let
[
begin{array}{l}
boldsymbol{A}=left[begin{array}{ccc}
2 boldsymbol{k}-mathbf{1} & mathbf{2} sqrt{boldsymbol{k}} & mathbf{2} sqrt{boldsymbol{k}} \
mathbf{2} sqrt{boldsymbol{k}} & mathbf{1} & -mathbf{2} boldsymbol{k} \
-mathbf{2} sqrt{boldsymbol{k}} & mathbf{2} boldsymbol{k} & mathbf{1}
end{array}right] \
boldsymbol{B}=left[begin{array}{ccc}
mathbf{0} & mathbf{2} boldsymbol{k}-mathbf{1} & sqrt{boldsymbol{k}} \
mathbf{1}-mathbf{2} boldsymbol{k} & mathbf{0} & mathbf{2} sqrt{boldsymbol{k}} \
-sqrt{boldsymbol{k}} & -mathbf{2} sqrt{boldsymbol{k}} & mathbf{0}
end{array}right] \
text { If det }(boldsymbol{A} boldsymbol{d} boldsymbol{j}(boldsymbol{A}))+operatorname{det}(boldsymbol{A} boldsymbol{d} boldsymbol{j}(boldsymbol{B}))=mathbf{1 0}^{boldsymbol{o}}
end{array}
]
then ( [k] ) is equal to
4
3.6
( c )
( D )
12
678 If a matrix is of order ( 2 times 3 ), then the
number of elements in the matrix is
A. 5
B. 6
( c cdot 2 )
D. 3
12
679 Find the inverse of the following matrices by the adjoining method ( left[begin{array}{cc}-1 & 5 \ -3 & 2end{array}right] ) 12
680 Construct a ( 2 times 3 ) matrix ( A=left[a_{i j}right] )
whose elements are given by ( a_{i j}= )
( left{begin{array}{l}i-j i geq j \ i+j i<jend{array}right. )
12
681 Two matrices are equal if and only if they have the and
corresponding elements are
A. rows, equal
B. order, equal
C . columns, equal
D. order, unequal
12
682 ( f(alpha, beta, gamma ) are three real numbers and
( boldsymbol{A}= )
( left[begin{array}{ccc}mathbf{1} & cos (boldsymbol{alpha}-boldsymbol{beta}) & cos (boldsymbol{alpha}-gamma) \ cos (boldsymbol{beta}-boldsymbol{alpha}) & mathbf{1} & cos (boldsymbol{beta}-gamma) \ cos (gamma-boldsymbol{alpha}) & cos (gamma-boldsymbol{beta}) & mathbf{1}end{array}right] )
then
This question has multiple correct options
A. ( A ) is symmetric
B. ( A ) is orthogona
c. ( A ) is singular
D. ( A ) is not invertible
12
683 ( fleft(begin{array}{lll}1 & 0 & 1 \ 0 & 1 & 2 \ 0 & 0 & 4end{array}right], ) then show that
[
|mathbf{3} boldsymbol{A}|=mathbf{2 7}|boldsymbol{A}|
]
12
684 5.
Let A=
0
0
(-1
0
-1
0
-1)
0 . The only correct
0
statement about the matrix A is
(a) A2 = 1
(b) A=(-1)1, where I is a unit matrix
(C) A-1 does not exist
(d) Ais a zero matrix
12
685 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{3} & mathbf{4}end{array}right], ) then find ( boldsymbol{A}+boldsymbol{A}^{boldsymbol{T}} ) 12
686 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{0} & mathbf{5} \ mathbf{0} & mathbf{0}end{array}right] ) and ( boldsymbol{f}(boldsymbol{x})=mathbf{1}+boldsymbol{x}+ )
( boldsymbol{x}^{2}+ldots ldots+boldsymbol{x}^{6}, operatorname{then} boldsymbol{f}(boldsymbol{A})= )
A . 0
в. ( left[begin{array}{ll}1 & 5 \ 0 & 1end{array}right] )
c. ( left[begin{array}{ll}1 & 5 \ 0 & 0end{array}right] )
D. ( left[begin{array}{ll}0 & 5 \ 1 & 1end{array}right] )
12
687 A matrix having ( m ) rows and ( n ) columns with ( m=n ) is said to be a
A. rectangular matrix
B. square matrix
c. identity matrix
D. scalar matrix
12
688 a
O
1
0
10.
If A =
and B=
, then value of a for which
5
1 |
(2003)
A2 =B, is
(a) 1
(c) 4
(b) -1
(d) no real values
12
689 If ( boldsymbol{A}=left[begin{array}{ll}1 & -2 \ 5 & -3end{array}right], ) then ( A+A^{T} ) equals
A. ( left[begin{array}{cc}2 & 3 \ 3 & -6end{array}right] )
В. ( left[begin{array}{cc}2 & -4 \ 10 & -6end{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}2 & 4 \ -10 & 6end{array}right] )
D. None of these
12
690 Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] )
whose elements are given by ( a_{i j}=2 i- )
( boldsymbol{j} )
12
691 Find the values of ( x, y ) and ( z ) if ( left[begin{array}{lll}boldsymbol{x} & mathbf{5} & mathbf{4} \ mathbf{5} & mathbf{9} & mathbf{1}end{array}right]=left[begin{array}{lll}mathbf{3} & mathbf{5} & boldsymbol{z} \ mathbf{5} & boldsymbol{y} & mathbf{1}end{array}right] ) 12
692 If
( A ) is a non-singular matrix, then
This question has multiple correct options
( mathbf{A} cdot A^{-1} ) is symmetric if ( A ) is symmeteric
B. ( A^{-1} ) is skew-symmetric if ( A ) is symmeteric
( mathbf{C} cdotleft|A^{-1}right|=|A| )
D ( cdotleft|A^{-1}right|=|A|^{-1} )
12
693 If ( A ) is a square matrix such that ( A^{2}= )
( I, ) then ( A^{-1} ) is equal to
A . ( I )
в. 0
( c . A )
( mathbf{D} cdot I+A )
12
694 Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] ) 12
695 If ( A=left[begin{array}{ll}1 & 2 \ 3 & 4end{array}right], B=left[begin{array}{ll}2 & 3 \ 4 & 5end{array}right], ) and ( 4 A-3 B+C )
( =0, ) then ( C= )
A. ( left[begin{array}{cc}2 & -1 \ 0 & 1end{array}right] )
В. ( left[begin{array}{cc}2 & 1 \ 0 & -1end{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}-2 & 1 \ 0 & -1end{array}right] )
D. None
12
696 If ( boldsymbol{A}=left[begin{array}{ccc}1 & 2 & -3 \ 5 & 0 & 2 \ 1-1 & 1end{array}right], B=left[begin{array}{ccc}3 & -12 \ 4 & 2 & 5 \ 2 & 0 & 3end{array}right] ) and
( boldsymbol{C}=left[begin{array}{ccc}mathbf{4} & mathbf{1} & mathbf{2} \ mathbf{0} & mathbf{3} & mathbf{2} \ mathbf{1}-mathbf{2 3}end{array}right], ) then compute ( (boldsymbol{A}+boldsymbol{B}) )
and ( (B-C) )
Also, verify that ( boldsymbol{A}+(boldsymbol{B}-boldsymbol{C})= )
( (boldsymbol{A}+boldsymbol{B})-boldsymbol{C} )
12
697 ( mathbf{f} boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{- 5} & mathbf{7} \ mathbf{0} & mathbf{7} & mathbf{9} \ mathbf{1 1} & mathbf{8} & mathbf{9}end{array}right], ) then trace of
matrix A is.
A . 17
B. 25
( c .3 )
D. 12
12
698 ( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{- 1} & mathbf{- 2} & mathbf{- 3}end{array}right], ) then A is a
nilpotent matrix of index
( A )
B. 3
( c cdot 4 )
D. 5
12
699 f ( boldsymbol{T} boldsymbol{r}(boldsymbol{A})=boldsymbol{6}, ) then ( boldsymbol{T} boldsymbol{r}(boldsymbol{4} boldsymbol{A})= )
( A cdot frac{3}{2} )
B. 2
c. 12
D. 24
12
700 Define a diagonal matrix. 12
701 Let ( M ) and ( N ) be two ( 3 times 3 ) be two non-
singular skew symmetric matrices
such that ( M N=N M ) further, if ( M neq )
( n^{2} ). If ( P T ) denotes the transpose of ( P ) then
( M^{2} N^{2}left(M^{T} Nright)^{-1}(M N)^{-1} T ) is equal to
12
702 ff ( 2left[begin{array}{ll}3 & 4 \ 5 & xend{array}right]+left[begin{array}{ll}1 & y \ 0 & 1end{array}right]=left[begin{array}{cc}7 & 0 \ 10 & 5end{array}right] )
matrices. Find the value of ( x & y )
12
703 If ( A ) and ( B ) are invertible matrices of
order
3. ( |boldsymbol{A}|=mathbf{2} ) and ( left|(boldsymbol{A B})^{-1}right|=-frac{mathbf{1}}{mathbf{6}} )
Find ( |boldsymbol{B}| )
12
704 If ( A ) be a ( 3 times 3 ) matrix and ( I ) be the unit
matrix of that order such that ( boldsymbol{A}= )
( A^{2}+I ) then ( A^{-1} ) is equal to
A . ( A )
в. ( A+I )
c. ( I-A )
D. ( A-I )
12
705 Find the transpose of matrix ( left[begin{array}{ll}2 & 5 \ 1 & 3end{array}right] ) 12
706 Determine the value of ( (x+y) ) if ( left[begin{array}{cc}2 x+y & 4 x \ 5 x-7 & 4 xend{array}right]=left[begin{array}{cc}7 & 7 y-12 \ y & x+6end{array}right] ) 12
707 If ( A ) and ( B ) are square matrices of the same order, explain, why in general? ( (A+B)^{2} neq A^{2}+2 A B+B^{2} ) 12
708 Let ( A ) and ( B ) be two symmetric matrices
of order 3
Statement-1:
( boldsymbol{A}(boldsymbol{B} boldsymbol{A}) ) and ( (boldsymbol{A B}) boldsymbol{A} ) are symmetric
matrices.
Statement-2:
( A B ) is symmetric matrix if matrix
multiplication of ( boldsymbol{A} ) and ( boldsymbol{B} ) is
commutative.
A. Statement-1 is true, Statement-2 is true; Statement-2 is correct explanation for Statement-1.
B. Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement- 1 .
c. Statement-1 is true, Statement-2 is false
D. Statement-1 is false, Statement-2 is true.
12
709 ( operatorname{Given} boldsymbol{A}=left[begin{array}{lll}mathbf{1} & mathbf{1} & mathbf{1} \ mathbf{2} & mathbf{4} & mathbf{1} \ mathbf{2} & mathbf{3} & mathbf{1}end{array}right], boldsymbol{B}=left[begin{array}{ll}mathbf{2} & mathbf{3} \ mathbf{3} & mathbf{4}end{array}right] )
Find ( P ) such that ( B P A=left[begin{array}{lll}1 & 0 & 1 \ 0 & 1 & 0end{array}right] )
12
710 ( mathrm{IF} boldsymbol{B}=left[begin{array}{cc}mathbf{1} & mathbf{3} \ -mathbf{2} & mathbf{5}end{array}right] ) and ( boldsymbol{C}=left[begin{array}{cc}-mathbf{2} & mathbf{5} \ mathbf{3} & mathbf{4}end{array}right] )
then find the value of ( B-4 C )
12
711 The element in the second row and third
column of the matrix ( left[begin{array}{ccc}mathbf{4} & mathbf{5} & mathbf{- 6} \ mathbf{3} & mathbf{- 4} & mathbf{3} \ mathbf{2} & mathbf{1} & mathbf{0}end{array}right] ) is
( A cdot 3 )
B.
( c cdot 2 )
D. –
12
712 The equation, ( left[begin{array}{ccc}mathbf{1} & boldsymbol{x} & boldsymbol{y}end{array}right]left[begin{array}{ccc}mathbf{1} & boldsymbol{3} & mathbf{1} \ mathbf{0} & boldsymbol{2} & -mathbf{1} \ mathbf{0} & mathbf{0} & mathbf{1}end{array}right]left[begin{array}{l}mathbf{1} \ boldsymbol{x} \ boldsymbol{y}end{array}right]=[mathbf{0}] ) has
for
(i) ( y=0 )
(p) rational
roots
(ii) ( y=-1 )
(q) irrational
roots
(r) integral roots
A ( . ) (i) ( (p) ) (ii) ( (r) )
B. (i) (q) (ii) (p)
( c cdot(i)(p)(text { ii) }(q) )
D. (i) (r) (ii) (p)
12
713 If the product of the matrices
[
left[begin{array}{ll}
1 & 1 \
0 & 1
end{array}right]left[begin{array}{ll}
1 & 2 \
0 & 1
end{array}right]left[begin{array}{ll}
1 & 3 \
0 & 1
end{array}right]
]
( left[begin{array}{cc}mathbf{1} & boldsymbol{n} \ mathbf{0} & mathbf{1}end{array}right]=left[begin{array}{cc}mathbf{1} & mathbf{3 7 8} \ mathbf{0} & mathbf{1}end{array}right], ) then ( boldsymbol{n} ) is
equal to
( A cdot 27 )
B . 26
( c .37 )
D. 37
12
714 If ( A ) is a skew-symmetric matrix and ( n ) is
a positive integer, then ( boldsymbol{A}^{boldsymbol{n}} ) is
A. a symmetric matrix
B. skew-symmetric matrix
c. diagonal matrix
D. none of these
12
715 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{0} & mathbf{0} \ mathbf{4} & mathbf{0}end{array}right], ) then find ( boldsymbol{A}^{16} ) 12
716 Find matrix ( boldsymbol{A} ) such that
( left[begin{array}{cc}2 & -1 \ 1 & 0 \ -3 & 4end{array}right] A=left[begin{array}{cc}-1 & -8 \ 1 & -2 \ 9 & 22end{array}right] )
12
717 ( mathbf{f} mathbf{A}=left[begin{array}{ccc}mathbf{0} & mathbf{1} & mathbf{4} \ mathbf{- 1} & mathbf{0} & mathbf{7} \ mathbf{- 4} & mathbf{- 7} & mathbf{0}end{array}right] ) then ( mathbf{A}^{mathbf{T}}= )
( A )
в. – А
( c )
D. ( A^{2} )
12
718 If ( A ) is a matrix of order ( 3 times 4 ) and ( B ) is a
matrix of order ( 4 times 3, ) find the order of
the matrix of ( boldsymbol{A B} )
12
719 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{3} & mathbf{7} \ mathbf{2} & mathbf{5}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{cc}-mathbf{3} & mathbf{2} \ mathbf{4} & -mathbf{1}end{array}right] ) find
the matrix ( C ) if ( 2 C=A+B )
12
720 Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] )
whose elements ( a_{i j} ) are given by:
( boldsymbol{a}_{boldsymbol{i} j}=left|frac{boldsymbol{3} boldsymbol{i}-boldsymbol{j}}{boldsymbol{2}}right| )
12
721 Find the values of ( x, y ) and ( z ) from the
matrix equation. ( left(begin{array}{cc}5 x+2 & y-4 \ 0 & 4 z+6end{array}right)=left(begin{array}{cc}12 & -8 \ 0 & 2end{array}right) )
12
722 ( mathbf{f} boldsymbol{A}=left[begin{array}{ccc}mathbf{1}^{2} & mathbf{2}^{mathbf{2}} & mathbf{3}^{2} \ mathbf{2}^{mathbf{2}} & mathbf{3}^{mathbf{2}} & mathbf{4}^{2} \ mathbf{3}^{mathbf{2}} & mathbf{4}^{mathbf{2}} & mathbf{5}^{mathbf{2}}end{array}right] ) then ( |boldsymbol{A} boldsymbol{d} boldsymbol{j} boldsymbol{A}|= )
( A )
B . 16
( c cdot 64 )
D. 128
12
723 Using elementary tranormtion, find the
[
text { inverse of }left[begin{array}{ccc}
mathbf{1} & mathbf{3} & -mathbf{2} \
-mathbf{3} & mathbf{0} & -mathbf{5} \
mathbf{2} & mathbf{5} & mathbf{0}
end{array}right]
]
12
724 Find the inverse of the following matrix
by using elementary row
transformation
( left[begin{array}{ccc}2 & 0 & -1 \ 5 & 1 & 0 \ 0 & 1 & 3end{array}right] )
12
725 If ( A=left[begin{array}{l}45 \ 21end{array}right], ) then show that ( A^{-1}= )
( frac{1}{6}(A-5 I) )
12
726 [1 2
32. If A = 2 1
a 2
2
-2 is a matrix satisfying the equation
6
AAT=91, where I is 3 x 3 identity matrix, then the ordered
pair (a, b) is equal to:
(JEE M 2015]
(a) (2,1)
(b) (-2,-1)
(c) (2, -1)
(d) (-2,1)
12
727 If ( boldsymbol{A}=left(begin{array}{ccc}mathbf{8} & mathbf{5} & mathbf{2} \ mathbf{1} & mathbf{- 3} & mathbf{4}end{array}right), ) then find ( boldsymbol{A}^{boldsymbol{T}} ) and
( left(A^{T}right)^{T} )
12
728 ( left[begin{array}{ccc}1 & 0 & 2 \ -1 & 1 & -2 \ 0 & 2 & 1end{array}right]+left[begin{array}{ccc}5 & 1 & -2 \ 1 & 1 & 0 \ -2 & -2 & 1end{array}right] )
What will be the sum of the diagonal
elements of the resultant matrix.
A . 10
B. 6
( c )
( D )
12
729 If ( A ) and ( B ) are two square matrix of
order ( n ) then prove that ( :(A B)^{-1}= )
( boldsymbol{B}^{-1} boldsymbol{A}^{-1} )
12
730 If ( A ) is a square matrix of order ( n ), then ( |boldsymbol{k} boldsymbol{A}|= )
A ( . k|A| )
в. ( k^{n}|A| )
c. ( k^{-n}|A| )
D. | ( A mid )
12
731 If ( boldsymbol{A}=left[begin{array}{ccc}-1 & 1 & -1 \ 3 & -3 & 3 \ 5 & 5 & 5end{array}right] ) and ( B= )
( left[begin{array}{ccc}0 & 4 & 3 \ 1 & -3 & -3 \ -1 & 4 & 4end{array}right], ) then find ( A^{2}-B^{2} )
12
732 ( mathbf{f}left[begin{array}{cc}boldsymbol{x}+mathbf{3} & mathbf{4} \ boldsymbol{y}-mathbf{4} & boldsymbol{x}+boldsymbol{y}end{array}right]=left[begin{array}{ll}mathbf{5} & mathbf{4} \ mathbf{3} & mathbf{9}end{array}right], ) find ( boldsymbol{x} ) and
( boldsymbol{y} )
12
733 Show that the elements on the main
diagonal of a skew-symmetric matrix are all zero.
12
734 ( left.begin{array}{l}text { Find } boldsymbol{a}, boldsymbol{b}, boldsymbol{c}, boldsymbol{d} text { if }left[begin{array}{cc}boldsymbol{d}+mathbf{1} & mathbf{1 0}+boldsymbol{a} \ mathbf{3} boldsymbol{b}-mathbf{2} & boldsymbol{a}-mathbf{4}end{array}right]= \ mathbf{2} quad mathbf{2} boldsymbol{a}+mathbf{1} \ boldsymbol{b}-mathbf{5} quad boldsymbol{4} boldsymbol{c}end{array}right] ) 12
735 If ( mathbf{0} leq[boldsymbol{x}]<mathbf{2},-mathbf{1} leq[boldsymbol{y}]<mathbf{1} ) and ( mathbf{1} leq )
( [z]<3([.] ) denotes the greatest integer
function) then the maximum value of
determinant
( boldsymbol{Delta}=left|begin{array}{ccc}{[boldsymbol{x}]+mathbf{1}} & {[boldsymbol{y}]} & {[boldsymbol{z}]} \ {[boldsymbol{x}]} & {[boldsymbol{y}]+mathbf{1}} & {[boldsymbol{z}]} \ {[boldsymbol{x}]} & {[boldsymbol{y}]} & {[boldsymbol{z}]+mathbf{1}}end{array}right| )
( A )
B. 2
( c cdot 3 )
( D )
12
736 Find Order of matrix : ( left[begin{array}{lll}mathbf{1} & mathbf{2} & mathbf{6} \ mathbf{2} & mathbf{4} & mathbf{3}end{array}right] ) 12
737 If ( A ) is a square matrix, then ( a d j A^{T}- )
( (a d j A)^{T} ) is equal to
A ( cdot 2|A| )
B . ( 2|A| I )
c. null matrix
D. unit matrix
12
738 ( boldsymbol{A}=left[begin{array}{ccc}mathbf{1} & mathbf{- 3} & mathbf{- 4} \ mathbf{- 1} & mathbf{3} & mathbf{4} \ mathbf{1} & mathbf{- 3} & mathbf{- 4}end{array}right] ) and ( mathbf{A}^{2}=boldsymbol{lambda} boldsymbol{I} ) then
( boldsymbol{lambda}= )
( A cdot 0 )
B.
( c cdot frac{1}{2} )
D. –
12
739 Let three matrices ( A=left[begin{array}{ll}2 & 1 \ 4 & 1end{array}right] ; B= )
( left[begin{array}{ll}mathbf{3} & mathbf{4} \ mathbf{2} & mathbf{3}end{array}right] ) and ( C=left[begin{array}{cc}mathbf{3} & -mathbf{4} \ -mathbf{2} & mathbf{3}end{array}right] ) then
( boldsymbol{t}_{boldsymbol{r}}(boldsymbol{A})+boldsymbol{t}_{boldsymbol{r}}left(frac{A B C}{2}right)+boldsymbol{t}_{boldsymbol{r}}left(frac{boldsymbol{A}(boldsymbol{B C})^{2}}{4}right)+ )
( boldsymbol{t}_{boldsymbol{r}}left(frac{boldsymbol{A}(boldsymbol{B} C)^{3}}{boldsymbol{8}}right)+ldots+infty )
( A cdot 6 )
B.
c. 12
D. none of these
12
740 If ( boldsymbol{A}=operatorname{diag}[mathbf{2},-mathbf{3},-mathbf{5}], boldsymbol{B}= )
( operatorname{diag}[4,-6,-3] ) and ( C=operatorname{diag}[-3,4,1] )
then find
( mathbf{2 A}+boldsymbol{B}-mathbf{5} boldsymbol{C} )
12
741 ff ( A=left[begin{array}{ccc}-1 & 0 & 0 \ 0 & x & 0 \ 0 & 0 & mend{array}right] ) is a scalar matrix
then ( boldsymbol{x}+boldsymbol{m}= )
( mathbf{A} cdot mathbf{0} )
B. –
( c cdot-2 )
D. -3
12
742 If ( A_{2 times 3}, B_{4 times 3} ) and ( C_{2 times 4} ) are three matrices then which of the following is/are defined?
A ( cdot A C^{T} B )
в. ( B^{T} C^{T} A )
c. ( A B^{T} C )
D. All of these
12
743 Find the vector equation of the plane
passing through points ( 4 i-3 j- )
( k, 3 i+7 j-10 k ) and ( 2 i+5 j-7 k ) and
show that the point ( i+2 j-3 k ) lies in
the plane.
12
744 If ( A, B ) are square matrices of order ( 3, A ) is non-singular and ( A B=O ), then ( B ) is
( a )
A. Null matrix
B. singular matrix
c. Unit matrix
D. Non-singular matrix
12
745 If ( l ) is an identity matrix and ( A ) is a
square matrix such that ( A^{2}=A ), then
find the value of ( (boldsymbol{l}+boldsymbol{A})^{2}-mathbf{3} boldsymbol{A} )
12
746 If ( D_{1} ) and ( D_{2} ) are two ( 3 times 3 ) diagonal
matrices, then
A. ( D_{1} D_{2} ) is a diagonal matrix
B. ( D_{1}+D_{2} ) is a diagonal matrix
c. ( D_{1}^{2}+D_{2}^{2} ) is a diagonal matrix
D. 1,2,3 are correct
12
747 Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] )
whose element ( a_{i j} ) is ( frac{(i+j)^{2}}{2} )
12
748 ( boldsymbol{A}=left[begin{array}{cc}cos boldsymbol{theta} & -sin boldsymbol{theta} \ sin boldsymbol{theta} & cos boldsymbol{theta}end{array}right] ) and ( boldsymbol{A B}=boldsymbol{B A}= )
( I, ) then ( B ) is equal to
( mathbf{A} cdotleft[begin{array}{cc}-cos theta & sin theta \ sin theta & cos thetaend{array}right] )
( mathbf{B} cdotleft[begin{array}{cc}cos theta & sin theta \ -sin theta & cos thetaend{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}-sin theta & cos theta \ cos theta & sin thetaend{array}right] )
( mathbf{D} cdotleft[begin{array}{cc}sin theta & -cos theta \ -cos theta & sin thetaend{array}right] )
12
749 ( boldsymbol{A}=left[begin{array}{cc}mathbf{2} & mathbf{3} \ mathbf{5} & mathbf{7}end{array}right] boldsymbol{B}=left[begin{array}{cc}mathbf{0} & mathbf{4} \ -mathbf{1} & mathbf{7}end{array}right] boldsymbol{C}= )
( left[begin{array}{cc}mathbf{1} & mathbf{0} \ mathbf{- 1} & mathbf{4}end{array}right] )
find ( boldsymbol{A} boldsymbol{C}+boldsymbol{B}^{2}-mathbf{1 0 C} )
12
750 If ( boldsymbol{A}=left[boldsymbol{a}_{i j}right]_{3 times 3} ) is a square matrix so
that ( a_{i j}=i^{2}-j^{2}, ) then ( A ) is a
A. unit matrix
B. symmetric marix
c. skew symmetric matrix
D. orthogonal matrix
12
751 ( fleft(begin{array}{cc}4 & 8 \ -2 & -4end{array}right] ) find ( A^{2} ) 12
752 ( mathbf{f} mathbf{A}=left{begin{array}{ll}mathbf{0} & mathbf{2} \ mathbf{3} & -mathbf{4}end{array}right}, mathbf{k} mathbf{A}=left{begin{array}{ll}mathbf{0} & mathbf{3} mathbf{a} \ mathbf{2} mathbf{b} & mathbf{2 4}end{array}right} )
then arrange the values of ( k, a, b, ) in
ascending order
( mathbf{A} cdot k, a, b )
в. ( b, a, k )
( mathbf{c} cdot a, k, b )
( mathbf{D} cdot b, k, a )
12
753 Given a square matrix ( boldsymbol{A}=left[boldsymbol{a}_{boldsymbol{g}}right], ) where
( a_{g}=hat{i}^{2}-hat{j}^{2} . ) Then matrix ( A ) is a unit
matrix or null matrix or a symmetric matrix a skew symmetric matrix. Select with a reason.
12
754 The matrix ( boldsymbol{B} ) is
A. Symmetric
B. Scalar
c. Skew hermitian
D. Skew- symmetric
12
755 Find the value of ( mathbf{x}, mathbf{y}, mathbf{z}left[begin{array}{cc}boldsymbol{x}+mathbf{2} & mathbf{6} \ mathbf{3} & mathbf{5} zend{array}right]= )
( left[begin{array}{cc}mathbf{3} & boldsymbol{y}^{2}+mathbf{4} \ mathbf{3} & mathbf{2 0}end{array}right] )
12
756 If ( boldsymbol{A}=left[boldsymbol{a}_{i j}right], boldsymbol{i}, boldsymbol{j}=1,2 ) where ( boldsymbol{a}_{i j} ) is
defined as ( a_{i j}=i^{2}+j^{2} ) then write the
sum of the elements of the matrix ( boldsymbol{A} )
12
757 If ( boldsymbol{A}=left[boldsymbol{a}_{boldsymbol{i j}}right] ) is a square matrix of even
order such that ( left[a_{i j}right]=i^{2}-j^{2}, ) then
A ( cdot A ) is a skew-symmetric matrix and ( |A|=0 )
B. ( A ) is symmetric matrix and |A| is a square
C. ( A ) is symmetric matrix and ( |A|=0 )
D. none of these
12
758 ( mathbf{f} boldsymbol{A}=left[begin{array}{lll}mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{0} & mathbf{5} & mathbf{7} \ mathbf{6} & mathbf{8} & mathbf{9}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{lll}mathbf{2} & mathbf{0} & mathbf{3} \ mathbf{3} & mathbf{0} & mathbf{5} \ mathbf{5} & mathbf{7} & mathbf{0}end{array}right] )
then find the value of ( 3 A-2 B )
12
759 If the matrix ( left[begin{array}{ccc}1 & 3 & lambda+2 \ 2 & 4 & 8 \ 3 & 5 & 10end{array}right] ) is singular
( operatorname{then} lambda= )
A . -2
B. 4
( c cdot 2 )
D. –
12
760 Find teh adjoint of the matrix ( boldsymbol{A}= ) ( left(begin{array}{ccc}1 & 0 & -1 \ 3 & 4 & 5 \ 0 & -6 & -7end{array}right) ) and hence find the
matrix ( boldsymbol{A}^{-1} )
12
761 If ( A ) and ( B ) are symmetric matrices and
( A B=B A, ) then ( A^{-1} B ) is a
A. Symmetric matrix
B. Skew-symmetric matrix
c. Identity matrix
D. None of these
12
762 If a ( M ) matrix ( A ) is such that ( A A^{T}= )
( boldsymbol{I}=boldsymbol{A}^{boldsymbol{T}} boldsymbol{A}, ) find ( |boldsymbol{A}|=? )
12
763 If order of matrix ( A ) is ( 4 times 3 ) and order of
matrix ( B ) is ( 3 times 5 ) then order of matrix
( boldsymbol{B}^{prime} boldsymbol{A}^{prime} ) is:
( mathbf{A} cdot 5 times 2 )
B. ( 4 times 5 )
( mathbf{c} cdot 5 times 4 )
D. ( 3 times 2 )
12
764 ( boldsymbol{A}=left[begin{array}{cc}boldsymbol{a} & boldsymbol{b} \ mathbf{0} & boldsymbol{c}end{array}right] ) then ( boldsymbol{A}^{-1}+(boldsymbol{A}-boldsymbol{a} boldsymbol{I})(boldsymbol{A}-1) )
( boldsymbol{c} boldsymbol{I})= )
A ( cdot frac{1}{a c}left[begin{array}{cc}a & b \ 0 & -cend{array}right] )
в. ( frac{1}{a c}left[begin{array}{cc}-a & b \ 0 & cend{array}right] )
c. ( frac{1}{a c}left[begin{array}{cc}c & -b \ 0 & aend{array}right] )
D ( cdot frac{1}{a c}left[begin{array}{cc}c & b \ 0 & aend{array}right] )
12
765 If ( boldsymbol{A}=left(begin{array}{l}832 \ 591end{array}right) ) and ( B=left(begin{array}{c}1-1 \ 0end{array}right) . ) Find ( A+B )
if it exists.
12
766 If ( A ) and ( B ) are square matrices of order
n’ such that ( A^{2}-B^{2}=(A-B)(A+ )
( B ) ), then which of the following will be
true?
A. Either of A or B is zero matrix
в. ( A=B )
c. ( A B=B A )
D. Either of A or B is an identity matrix
12
767 If ( A^{-1}=left[begin{array}{ccc}3 & -1 & 1 \ -15 & 6 & -5 \ 5 & -2 & 2end{array}right] ) and ( B= )
( left[begin{array}{ccc}1 & 2 & -2 \ -1 & 3 & 0 \ 0 & -2 & 1end{array}right] ), find ( (A B)^{-1} )
12
768 Using elementary transformation, find the inverse of the matrix ( boldsymbol{A}= ) ( left[begin{array}{cc}boldsymbol{a} & boldsymbol{b} \ boldsymbol{c} & left(frac{1+boldsymbol{b} c}{boldsymbol{a}}right)end{array}right] )
( ^{mathbf{A}} rightarrow A^{-1}=left[begin{array}{cc}frac{1+b c}{a} & b \ -c & aend{array}right] )
( stackrel{mathbf{B}}{rightarrow} Rightarrow A^{-1}=left[begin{array}{cc}frac{1+b c}{a} & -b \ c & aend{array}right] )
( stackrel{mathrm{c}}{*} quad, A^{-1}=left[begin{array}{cc}frac{1+b c}{a} & b \ c & aend{array}right] )
D. None of these.
12
769 Is this possible ( a neq )
( mathbf{0} cdotleft|begin{array}{ccc}boldsymbol{x}+mathbf{1} & boldsymbol{x} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x}+boldsymbol{a} & boldsymbol{x} \ boldsymbol{x} & boldsymbol{x} & boldsymbol{x}+boldsymbol{a}^{2}end{array}right|=mathbf{0} )
represents a straight line parallel to the
y-axis
A. True
B. False
12
770 If ( boldsymbol{A}^{T} boldsymbol{B}^{T}=boldsymbol{C}^{T} ) then ( mathbf{C}= )
( A cdot A B )
B. BA
( c . ) вс
D. ABC
12
771 [
operatorname{ftg}left[begin{array}{ccc}
3 & 2 & -1 \
2 & -2 & 0 \
1 & 3 & 1
end{array}right], Bleft[begin{array}{ccc}
-3 & -1 & 0 \
2 & 1 & 3 \
4 & -1 & 2
end{array}right]
]
and ( X=A+B ) then find ( X )
12
772 The sum and product of matrices A and B exist. Which of the
following implications are necessarily true?
1. A and B are square matrices of
same order.
2. A and B are non-singular matrices.
Select the correct answer using the
code given below:
A . 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2
12
773 Find the inverse of the following matrix by using elementary row transformation:
( left[begin{array}{ll}5 & 2 \ 2 & 1end{array}right] )
12
774 For two ( 3 times 3 ) matrices ( A ) and ( B ), let
( A+B=2 B^{prime} ) and ( 3 A+2 B=I_{3}, ) where
( B^{prime} ) is the transpose of ( B ) and ( I_{3} ) is ( 3 times 3 )
identify matrix. Then :
A. ( 5 A+10 B=2 I_{3} )
В . ( 3 A+6 B=2 I_{3} )
c. ( 10 A+5 B=3 I_{3} )
D. ( B+2 A=I_{3} )
12
775 If ( boldsymbol{A}=left[begin{array}{cc}cos boldsymbol{x} & sin boldsymbol{x} \ -sin boldsymbol{x} & cos boldsymbol{x}end{array}right], ) show that
( A^{2}= )
( left[begin{array}{cc}cos 2 x & sin 2 x \ -sin 2 x & cos 2 xend{array}right] ) and ( A^{prime} A=I )
12
776 If ( boldsymbol{A}=left[begin{array}{ccc}0 & 1 & 2 \ 1 & 2 & 3 \ 2 & 3 & 4end{array}right] ) and ( B=left[begin{array}{cc}1 & -2 \ -1 & 0 \ 2 & -1end{array}right] )
Check ( A B=B A ? )
12
777 If ( boldsymbol{A}=left[begin{array}{lll}mathbf{0} & mathbf{0} & mathbf{1} \ mathbf{0} & mathbf{1} & mathbf{0} \ mathbf{1} & mathbf{0} & mathbf{0}end{array}right], ) then ( boldsymbol{A}^{-1} ) is
A . ( -A )
в.
( c cdot 1 )
D. None of these
12
778 The order of ( [x, y, z]left[begin{array}{lll}a & h & g \ h & b & f \ g & f & cend{array}right]left[begin{array}{l}x \ y \ zend{array}right] ) is
( A cdot 3 times 1 )
B. ( 1 times 1 )
c. ( 1 times 3 )
D. ( 3 times 3 )
12
779 If ( A ) is a scalar matrix ( k I ) with scalar
( k neq 0 ) of order ( 3, ) the ( A^{-1} ) is:
A ( cdot frac{1}{k^{2}} I )
B. ( frac{1}{k^{3}} )
c. ( frac{1}{k} I )
D. ( k I )
12
780 Find the inverse of ( left[begin{array}{ccc}3 & -1 & -2 \ 2 & 0 & -1 \ 3 & -5 & 0end{array}right] )
using elementary row transformations.
12
781 If ( operatorname{Tr}(mathbf{A})=mathbf{2}+mathbf{i}, ) Then ( operatorname{Tr}[(mathbf{2}-mathbf{i}) mathbf{A}]= )
A ( .2+i )
в. ( 2-i )
( c .3 )
D. 5
12
782 The numbers of ( 3 times 3 ) matrices A whose
entries are either 0 or 1 and for which the system ( boldsymbol{A}left[begin{array}{l}boldsymbol{x} \ boldsymbol{y} \ boldsymbol{z}end{array}right]=left[begin{array}{l}mathbf{1} \ mathbf{0} \ mathbf{0}end{array}right] ) has exactly
two distinct solutions is?
A .
B . ( 2^{9}-1 )
( c .168 )
D.
12
783 If ( A_{2 times 3}, B_{4 times 3}, C_{2 times 4} ) are three matrices,
then which of the following is/are defined ?
A ( . A C^{T} B )
B . ( B^{T} C^{T} A )
( mathbf{c} cdot A B^{T} C )
D. All of these
12
784 If ( boldsymbol{A}=left[begin{array}{c}-1 \ 2 \ mathbf{3}end{array}right], boldsymbol{B}=[-mathbf{2},-mathbf{1},-mathbf{4}] ) verify
that ( (boldsymbol{A B})^{T}=boldsymbol{B}^{boldsymbol{T}} boldsymbol{A}^{boldsymbol{T}} )
12
785 Let ( boldsymbol{A}=left[begin{array}{ll}mathbf{2} & mathbf{4} \ mathbf{3} & mathbf{2}end{array}right], boldsymbol{B}=left[begin{array}{cc}mathbf{1} & mathbf{3} \ -mathbf{2} & mathbf{5}end{array}right] ) and
( boldsymbol{C}=left[begin{array}{cc}mathbf{- 2} & mathbf{5} \ mathbf{3} & mathbf{4}end{array}right] . ) Find:
( boldsymbol{B}-mathbf{4} boldsymbol{C} )
12
786 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{0} & mathbf{1} \ mathbf{1} & mathbf{0}end{array}right], ) then ( boldsymbol{A}^{4}= )
A. ( left[begin{array}{ll}1 & 0 \ 0 & 1end{array}right] )
в. ( left[begin{array}{ll}1 & 1 \ 0 & 0end{array}right] )
c. ( left[begin{array}{ll}0 & 0 \ 1 & 1end{array}right] )
D. ( left[begin{array}{ll}0 & 1 \ 1 & 0end{array}right] )
12
787 If two square matrices ( A ) and ( B ) are of same order and, ( operatorname{Tr}(boldsymbol{A})=mathbf{3}, boldsymbol{T} boldsymbol{r}(boldsymbol{B})=mathbf{5} )
( operatorname{then} operatorname{Tr}(boldsymbol{A}+boldsymbol{B})= )
A . 15
B. 8
( c cdot 3 / 5 )
D. cannot be determined
12
788 If the traces of the matrices ( A ) and ( B ) are
20 and ( 8, ) then trace of ( mathbf{A}+mathbf{B}= )
A . 28
B. 20
( c cdot-8 )
D. 12
12
789 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{3} & mathbf{1} \ mathbf{7} & mathbf{5}end{array}right], ) find the values of ( boldsymbol{x} ) and
( y ) such that ( A^{2}+x I_{2}=y A )
12
790 If ( P ) is a ( 3 times 3 ) matrix such that ( P^{T}= )
( 2 P+I ) where ( P^{T} ) is the transpose of ( P )
and lis the ( 3 times 3 ) identify matrix, then there exists a column matrix ( boldsymbol{X}= ) ( left[begin{array}{l}x \ y \ zend{array}right] neqleft[begin{array}{l}0 \ 0 \ 0end{array}right] ) such that
( ^{mathbf{A}} cdot operatorname{IX}=left[begin{array}{l}0 \ 0 \ 0end{array}right] )
B. PX = X
( c cdot ) Рх ( =2 x )
D. PX = – x
12
791 If ( A=left|begin{array}{cc}2 & -3 \ 3 & 2end{array}right| ) and ( B=left|begin{array}{cc}3 & -2 \ 2 & 3end{array}right| ) then ( 2 A-B= )
A. 11
( begin{array}{ll}1 & 4 \ 4 & 1end{array} )
в. ( mid begin{array}{ll}1 & 4 \ 1 & 4end{array} )
c. ( left|begin{array}{cc}1 & -4 \ 4 & 1end{array}right| )
D. ( mid begin{array}{ll}4 & 1 \ 1 & 4end{array} )
12
792 Solve for ( x ) and ( y ) if ( left(begin{array}{l}2 x+y \ x-3 yend{array}right)=left(begin{array}{c}5 \ 13end{array}right) ) 12
793 The number of nonzero diagonal
matrices of order 3 satisfying ( A^{2}=A ) is
12
794 Using elementary row transformations, find the inverse of the matrix ( boldsymbol{A}= )
( left[begin{array}{ccc}1 & 2 & 3 \ 2 & 5 & 7 \ -2 & -4 & -5end{array}right] )
12
795 If ( A ) and ( B ) are matrices given below:
[
begin{array}{l}
A=left[begin{array}{ccc}
0 & c & -b \
-c & o & a \
b & -a & 0
end{array}right] text { and } B= \
{left[begin{array}{ccc}
a^{2} & a b & a c \
a b & b^{2} & b c \
a c & b c & c^{2}
end{array}right]}
end{array}
]
then ( A B ) is a unit matrix. Is this
statement true?
12
796 If ( boldsymbol{A}=left[boldsymbol{a}_{i j}right]_{boldsymbol{m} times boldsymbol{n}^{prime}} boldsymbol{B}=left[boldsymbol{b}_{boldsymbol{i} j}right]_{boldsymbol{m} times boldsymbol{n}^{prime}} ) then the
element ( C_{23} ) of the matrix ( C=A+B )
is:
A ( . C_{23} )
В. ( a_{23}+b_{32} )
( mathbf{c} cdot a_{23}+b_{23} )
D ( cdot a_{32}+b_{23} )
12
797 The transpose of a row matrixis
A. zero matrix
B. diagonal matrix
C. column matrix
D. row matrix
12
798 Write the following as a single matrix ( left[begin{array}{cc}-1 & 2 \ 1 & -2 \ 3 & -1end{array}right]+left[begin{array}{cc}0 & 1 \ -1 & 0 \ -2 & 1end{array}right] ) 12
799 Find the inverse of the following matrix by using elementary row transformation:
( left[begin{array}{cc}mathbf{1} & mathbf{2} \ mathbf{2} & -mathbf{1}end{array}right] )
12
800 If ( boldsymbol{A}=left[begin{array}{cc}cos boldsymbol{alpha} & sin boldsymbol{alpha} \ -sin boldsymbol{alpha} & cos boldsymbol{alpha}end{array}right], ) then verify that
( boldsymbol{A}^{boldsymbol{T}} boldsymbol{A}=boldsymbol{I}_{2} )
12
801 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{1} & -mathbf{2} \ mathbf{3} & mathbf{0}end{array}right], boldsymbol{B}=left[begin{array}{cc}mathbf{- 1} & mathbf{4} \ mathbf{2} & mathbf{3}end{array}right], boldsymbol{C}= )
( left[begin{array}{cc}mathbf{0} & mathbf{1} \ mathbf{- 1} & mathbf{0}end{array}right], ) then ( mathbf{5} boldsymbol{A}-mathbf{3} boldsymbol{B}+mathbf{2} boldsymbol{C}= )
( mathbf{A} cdotleft[begin{array}{cc}8 & 20 \ 7 & 9end{array}right] )
( mathbf{B} cdotleft[begin{array}{cc}8 & -20 \ 7 & -9end{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}-8 & 20 \ -7 & 9end{array}right] )
( mathbf{D} cdotleft[begin{array}{cc}8 & 7 \ -20 & -9end{array}right] )
12
802 Find ( boldsymbol{X} ) and ( boldsymbol{Y}, ) if ( boldsymbol{2}left[begin{array}{ll}mathbf{1} & boldsymbol{3} \ boldsymbol{0} & boldsymbol{x}end{array}right]+left[begin{array}{ll}boldsymbol{y} & boldsymbol{0} \ boldsymbol{1} & boldsymbol{2}end{array}right]= )
( left[begin{array}{ll}5 & 6 \ 1 & 8end{array}right] )
12
803 For 3 x 3 matrices M and N, which of the following
statement(s) is (are) NOT correct? (JEE Adv. 2013)
(a) N’MN is symmetric or skew symmetric, according as
Mis symmetric or skew symmetric
(b) MN-NM is skew symmetric for all symmetric matrices
M and N
(c) MN is symmetric for all symmetric matrices Mand N
(d) (adj M) (adj N)= adj (MN) for all invertible matrices M
and N
12
804 atrices such
26. Let A = 2
1 0. Ifu, and tl, are column matrices
3
2
1
0)
that Auy = 0 and Au2 = 1 , then u + uz is equal to :
[2012
(a)
12
805 If ( M ) is a ( 3 times 3 ) matrix, where ( M^{T} M= )
( boldsymbol{I} ) and ( operatorname{det}(boldsymbol{M})=mathbf{1} ) then prove that
( operatorname{det}(M-I)=0 )
12
806 Find the inverse of the following matrix using transformation method. ( left[begin{array}{cc}mathbf{1} & mathbf{2} \ mathbf{2} & -mathbf{1}end{array}right] ) 12
807 ( operatorname{Given} boldsymbol{F}(boldsymbol{x})=left[begin{array}{ccc}cos boldsymbol{x} & -sin boldsymbol{x} & mathbf{0} \ sin boldsymbol{x} & cos boldsymbol{x} & boldsymbol{0} \ boldsymbol{0} & boldsymbol{0} & boldsymbol{1}end{array}right] )
( boldsymbol{x} in boldsymbol{R} ) Then prove ( boldsymbol{y}, boldsymbol{F}(boldsymbol{x}+boldsymbol{y})= )
( boldsymbol{F}(boldsymbol{x}) boldsymbol{F}(boldsymbol{y}) )
12
808 Assertion
If ( A ) is a square matrix of order ( n ) then
( operatorname{det}(k A)=k^{n}|A| )
Reason
If matrix ( mathrm{B} ) is obtained from ( mathrm{A} ) by multiplying any row (or column) by a
non zero scalar ( k ) then ( operatorname{det}(B)= )
( boldsymbol{k} operatorname{det}(boldsymbol{A}) )
A. Both (A) & (R) are individually true & (R) is correct explanation of ( (A) )
B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A)
c. (A)is true but (R) is false,
D. (A)is false but (R) is true.
12
809 If ( boldsymbol{A}=left[begin{array}{cc}mathbf{3} & mathbf{1} \ -mathbf{1} & mathbf{2}end{array}right], ) show that ( boldsymbol{A}^{2}-mathbf{5} boldsymbol{A}+ )
( mathbf{7} boldsymbol{I}=boldsymbol{O} . ) Hence find ( boldsymbol{A}^{-1} )
12
810 Let ( A ) and ( B ) be square matrices of the
other ( 3 times 3 . ) Is ( (A B)^{2}=A^{2} B^{2} ? ) Give
reasons.
12
811 ff ( left[begin{array}{cc}mathbf{4} & -mathbf{3} \ mathbf{2} & mathbf{1 6}end{array}right]=left[begin{array}{cc}mathbf{4} & -mathbf{3} \ mathbf{2} & mathbf{2}^{t}end{array}right], ) then ( mathbf{t}= )
( A cdot 2 )
B. 3
( c cdot 4 )
( D )
12
812 If ( A ) is a skew symmetric matrix of order
3, then the value of ( |boldsymbol{A}| ) is
A . 3
B. 0
( c .9 )
D. 27
12
813 ( fleft(begin{array}{ccc}2 & 3 & 4 \ 5 & -3 & 8 \ 9 & 2 & 16end{array}right], ) then trace of ( A ) is
A . 17
B. 25
( c cdot 8 )
D. 15
12
814 Inverse of ( boldsymbol{A}=left[begin{array}{cc}mathbf{1} & mathbf{3} \ mathbf{2} & -mathbf{2}end{array}right] ) is equal to? ( mathbf{A} )
( mathbf{A} cdot-frac{1}{8}left[begin{array}{cc}3 & 1 \ -2 & 2end{array}right] )
В. ( -frac{1}{8}left[begin{array}{rr}-2 & -3 \ -2 & 1end{array}right] )
( ^{mathbf{c}} cdot frac{1}{8}left[begin{array}{cc}-1 & -3 \ -2 & 2end{array}right] )
D. None of these
12
815 The inverse of ( left[begin{array}{lll}1 & a & b \ 0 & x & 0 \ 0 & 0 & 1end{array}right] ) is
( left[begin{array}{ccc}1 & -a & -b \ 0 & 1 & 0 \ 0 & 0 & 1end{array}right] ) then ( x= )
( A )
в.
( c .0 )
( D )
12
816 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{3} & -mathbf{2} \ mathbf{4} & -mathbf{2}end{array}right], ) find ( boldsymbol{K} ) such that ( boldsymbol{A}^{mathbf{2}}= )
( boldsymbol{K} boldsymbol{A}-boldsymbol{2} boldsymbol{I}, ) where ( boldsymbol{I} ) is the identity
element.
12
817 ( left[begin{array}{ll}1 & -tan theta \ tan theta & 1end{array}right]left[begin{array}{ll}1 & tan theta \ -tan theta & 1end{array}right] )
( =left[begin{array}{ll}a & -b \ -b & aend{array}right] ) then
( A cdot a=1 )
B ( cdot a=sec ^{2} theta, b=0 )
c. ( a=0, b=sin ^{2} theta )
( mathbf{D} cdot mathbf{a}=sin 2 theta, mathbf{b}=cos 2 theta )
12
818 Let ( A=left(begin{array}{l}3^{2} \ 5end{array}right) ) and ( B=left(begin{array}{c}8-1 \ 3end{array}right) . ) Find the
matrix ( mathrm{C} ) if ( boldsymbol{C}=mathbf{2} boldsymbol{A}+boldsymbol{B} )
12
819 Find the inverse of the following matrix by using elementary row transformation
( left[begin{array}{cc}mathbf{3} & mathbf{1 0} \ mathbf{2} & mathbf{7}end{array}right] )
12
820 IIf ( mathbf{A}=left[begin{array}{ll}boldsymbol{a} & mathbf{0} \ boldsymbol{a} & mathbf{0}end{array}right], mathbf{B}=left[begin{array}{ll}mathbf{0} & mathbf{0} \ boldsymbol{b} & boldsymbol{b}end{array}right], ) then
( mathbf{A B}= )
( mathbf{A} cdot mathbf{0} )
в. ВА
c. АВ
D. ABAB
12
821 ( mathrm{If} mathrm{A}+mathrm{B}=left[begin{array}{lll}1 & 0 & 2 \ 2 & 2 & 2 \ 1 & 1 & 1end{array}right] ) and ( A-B= )
( left[begin{array}{ccc}1 & 4 & 4 \ 4 & 2 & 0 \ -1 & -1 & 2end{array}right] )
then prove that ( A=left[begin{array}{lll}1 & 2 & 3 \ 3 & 2 & 1 \ 0 & 0 & 2end{array}right] ) and ( B= )
( left[begin{array}{ccc}0 & -2 & -1 \ -1 & 0 & 1 \ 1 & 1 & 0end{array}right] )
12
822 ( operatorname{Let} boldsymbol{A}=left(begin{array}{cc}mathbf{3} & mathbf{2} \ mathbf{5} & mathbf{1}end{array}right) ) and ( boldsymbol{B}=left(begin{array}{cc}mathbf{8} & -mathbf{1} \ mathbf{4} & mathbf{3}end{array}right) )
Find the matrix ( C, ) if ( C=2 A+B )
12
823 ( (boldsymbol{A}+boldsymbol{B})^{boldsymbol{T}}= )
( A cdot A+B )
В. ( A^{T}+B^{T} )
c. Does not exist
D. (a) or (b)
12
824 Solve the equation for ( x, y, z ) and ( t ) if ( mathbf{2}left[begin{array}{ll}boldsymbol{x} & boldsymbol{z} \ boldsymbol{y} & boldsymbol{t}end{array}right]+mathbf{3}left[begin{array}{cc}mathbf{1} & -mathbf{1} \ mathbf{0} & mathbf{2}end{array}right]=mathbf{3}left[begin{array}{ll}mathbf{3} & mathbf{5} \ mathbf{4} & mathbf{6}end{array}right] ) 12
825 Show that square matrix ( A ) and its
transpose ( A^{T} ) have the same eigen
values.
12
826 If ( A ) is a skew-symmetric matrix and ( n )
is odd positive integer, then ( A^{n} ) is
A. a skew-symmetric matrix
B. a symmetric matrix
c. a diagonal matrix
D. none of these
12
827 Find the inverse of the following matrices by the adjoining method
[
left[begin{array}{lll}
1 & 2 & 3 \
0 & 2 & 4 \
0 & 0 & 5
end{array}right]
]
12
828 Assertion If ( boldsymbol{A}=left[begin{array}{cc}mathbf{1} & boldsymbol{pi} \ mathbf{0} & mathbf{1}end{array}right], ) then ( boldsymbol{A}^{100}=left[begin{array}{cc}mathbf{1} & mathbf{1 0 0} boldsymbol{pi} \ mathbf{0} & mathbf{1}end{array}right] )
Reason
If ( B ) is a ( 2 times 2 ) matrix such that ( B^{2}=0 )
then ( (I+B)^{n}=I+n B ) for each ( n in )
( N )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
829 Write the number of all possible matrices of order ( 2 times 2 ) with each entry
1,2 or 3
12
830 ( boldsymbol{A}=left(begin{array}{cc}mathbf{5} & mathbf{2} \ mathbf{7} & mathbf{3}end{array}right) ) and ( boldsymbol{B}=left(begin{array}{cc}mathbf{2} & mathbf{- 1} \ mathbf{- 1} & mathbf{1}end{array}right) )
verify that ( (boldsymbol{A B})^{boldsymbol{T}}=boldsymbol{B}^{boldsymbol{T}} boldsymbol{A}^{boldsymbol{T}} )
12
831 If ( boldsymbol{A}=left[begin{array}{rr}mathbf{2} & -mathbf{3} \ -mathbf{4} & mathbf{1}end{array}right], ) then ad
( left(3 A^{2}+12 Aright) ) is equal to:
A. ( left[begin{array}{ll}51 & 63 \ 84 & 72end{array}right] )
B. ( left[begin{array}{ll}51 & 84 \ 63 & 72end{array}right] )
c. ( left[begin{array}{cc}72 & -63 \ -84 & 51end{array}right] )
D. ( left[begin{array}{cc}72 & -84 \ -63 & 51end{array}right] )
12
832 Two matrices ( A ) and ( B ) are added if
A. both are rectangular
B. both have same order
C. no of columns of ( A ) is equal to columns of ( B )
D. no of rows of A is equal to no of columns of B
12
833 ( mathbf{f}left[begin{array}{rr}boldsymbol{a}+mathbf{4} & mathbf{3} boldsymbol{b} \ mathbf{8} & -mathbf{6}end{array}right]=left[begin{array}{cc}mathbf{2} boldsymbol{a}+mathbf{2} & boldsymbol{b}+mathbf{2} \ boldsymbol{8} & boldsymbol{a}-mathbf{8} boldsymbol{b}end{array}right] )
then write the value of ( a-2 b )
12
834 Find ( boldsymbol{x} ) and ( boldsymbol{y}, ) when ( boldsymbol{x}+boldsymbol{y}=left[begin{array}{cc}mathbf{7} & mathbf{0} \ mathbf{2} & mathbf{5}end{array}right] )
and ( boldsymbol{x}-boldsymbol{y}=left[begin{array}{cc}mathbf{3} & mathbf{0} \ mathbf{0} & mathbf{3}end{array}right] )
12
835 Find the inverse of the following matrix by using elementary row transformation
( left[begin{array}{lll}0 & 1 & 2 \ 1 & 2 & 3 \ 3 & 1 & 1end{array}right] )
12
836 ( mathbf{f}left[begin{array}{ccc}mathbf{9} & -mathbf{1} & mathbf{4} \ -mathbf{2} & mathbf{1} & mathbf{3}end{array}right]=boldsymbol{A}+left[begin{array}{ccc}mathbf{1} & mathbf{2} & -mathbf{1} \ mathbf{0} & mathbf{4} & mathbf{9}end{array}right] )
then find the matrix ( A )
12
837 ff ( left[begin{array}{cc}2 & -1 \ 2 & 0end{array}right]+2 A=left[begin{array}{cc}-3 & 5 \ 4 & 3end{array}right], ) then the
matrix A equals
A. ( left[begin{array}{ll}-5 & 6 \ 2 & 3end{array}right] )
B. ( left[begin{array}{cc}-frac{5}{2} & 3 \ 1 & frac{3}{2}end{array}right] )
( mathbf{c} cdotleft[begin{array}{cc}-frac{5}{2} & 6 \ 2 & 3end{array}right] )
D. ( left[begin{array}{cc}-5 & 8 \ 1 & 3end{array}right] )
12
838 If ( A ) is ( 3 times 4 ) matrix and ( B ) is matrix such
that ( A^{prime} B ) and ( B A^{prime} ) are both defined, then ( B )
is of the type.
( A cdot 3 times 4 )
B. ( 3 times 3 )
( mathbf{c} cdot 4 times 4 )
D. ( 4 times 3 )
12
839 If ( A ) is a real skew-symmetric matrix
such that ( A^{2}+I=O, ) then
( mathbf{A} cdot A ) is a square matrix of even order with ( |A|=pm 1 )
B. ( A ) is a square matrix of odd order with ( |A|=pm 1 )
C ( cdot A ) can be a square matrix of any order with ( |A|=pm 1 )
D. ( A ) is a skew-symmetric matrix of even order with
( |A|=1 )
12
840 If ( boldsymbol{A}=left[begin{array}{lll}mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{3} & mathbf{2} & mathbf{1}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{lll}mathbf{3} & mathbf{2} & mathbf{1} \ mathbf{1} & mathbf{2} & mathbf{3}end{array}right] )
find ( mathbf{3} boldsymbol{B}-mathbf{2} boldsymbol{A} )
12
841 Let ( A ) be a ( 2 times 2 ) matrix and ( B=A+ )
( A^{T} . ) Then show that ( B ) is a symmetric
matrix.
12
842 If ( A ) is a non zero square matrix of order
( boldsymbol{n} ) with ( operatorname{det}(boldsymbol{I}+boldsymbol{A}) neq mathbf{0}, ) and ( boldsymbol{A}^{3}=mathbf{0} )
where ( I, O ) are unit and null matrices of
order ( n times n ) respectively, then
( (boldsymbol{I}+boldsymbol{A})^{-1}= )
A ( cdot I-A+A^{2} )
B . ( I+A+A^{2} )
c. ( I+A^{2} )
( mathbf{D} cdot I+A )
12
843 Let ( A ) being a square matrix, then prove that ( boldsymbol{A}+boldsymbol{A}^{T} ) is symmetric. 12
844 If ( boldsymbol{A}=left(begin{array}{cc}1 & -2 \ -mathbf{3} & 4end{array}right) ) and ( boldsymbol{A}+boldsymbol{B}=boldsymbol{O}, ) then ( mathbf{B} ) is
A ( cdotleft(begin{array}{cc}1 & -2 \ -3 & 4end{array}right) )
B. ( left(begin{array}{c}-1 \ 3end{array}right) )
c. ( left(begin{array}{cc}-1 & -2 \ -3-4end{array}right) )
D. ( left(begin{array}{c}1 \ 0 \ 0end{array}right) )
12
845 Determine whether the product of the matrices is defined in each case. If ( s 0 )
state the order of the product. MN,
where ( boldsymbol{M}=left[boldsymbol{m}_{i j}right]_{3 times 1}, boldsymbol{N}=left[boldsymbol{n}_{i j}right]_{1 times 5} )
12
846 Find the value of ( x, y, z ) if ( left[begin{array}{ll}mathbf{4} & mathbf{3} \ boldsymbol{x} & mathbf{5}end{array}right]=left[begin{array}{ll}boldsymbol{y} & boldsymbol{z} \ mathbf{1} & mathbf{5}end{array}right] ) 12
847 If ( boldsymbol{A}=left[begin{array}{cc}boldsymbol{4} & boldsymbol{3} \ boldsymbol{1} & boldsymbol{2}end{array}right] boldsymbol{B}=left[begin{array}{ll}boldsymbol{2} & boldsymbol{1} \ boldsymbol{1} & boldsymbol{2}end{array}right] )
Verify ( (boldsymbol{A} boldsymbol{B})^{prime}=boldsymbol{B}^{prime} boldsymbol{A}^{prime} )
12
848 If ( A=left[begin{array}{ccc}2 & 3 & 1 \ 0 & -1 & 5end{array}right], B=left[begin{array}{ccc}1 & 2 & -1 \ 0 & -1 & 3end{array}right] ) 12
849 If ( [A] neq 0 ) then which of the following is
not true?
A ( cdotleft(A^{2}right)^{-1}=left(A^{-1}right)^{2} )
B. ( left(A^{prime}right)^{-1}=left(A^{-1}right)^{prime} )
( mathbf{c} cdot A^{-1}=|A|^{-1} )
D. None of these
12
850 Find ( boldsymbol{A}^{boldsymbol{T}}: )
( boldsymbol{A}=left[begin{array}{lll}mathbf{4} & mathbf{3} & mathbf{- 1} \ mathbf{6} & mathbf{8} & mathbf{- 3} \ mathbf{4} & mathbf{1} & mathbf{3}end{array}right] )
12
851 If ( A ) is ( 4 times 5 ) matrix, if ( A^{T} B ) and ( B A^{T} )
are defined then ( B= )
( mathbf{A} cdot 5 times 4 )
B. ( 4 times 4 )
( mathbf{c} .5 times 5 )
D. ( 4 times 5 )
12
852 ( mathbf{f} mathbf{A}=left[begin{array}{lll}mathbf{1} & mathbf{- 3} & mathbf{- 4} \ mathbf{- 1} & mathbf{3} & mathbf{4} \ mathbf{1} & mathbf{- 3} & mathbf{- 4}end{array}right], ) then ( mathbf{A}^{2}= )
( A cdot A )
в. -4
c. Null matrix
D. ( 2 A )
12
853 If ( boldsymbol{m}left[begin{array}{ll}-mathbf{3} & mathbf{4}end{array}right]+boldsymbol{n}left[begin{array}{ll}mathbf{4} & -mathbf{3}end{array}right]=left[begin{array}{ll}mathbf{1 0} & -mathbf{1 1}end{array}right] )
then ( 3 m+7 n= )
( A cdot 3 )
B. 5
c. 10
( D )
12
854 ( mathbf{f} boldsymbol{A}=left[begin{array}{lll}mathbf{3} & mathbf{2} & mathbf{0} \ mathbf{1} & mathbf{4} & mathbf{0} \ mathbf{0} & mathbf{0} & mathbf{5}end{array}right] . ) Show that ( boldsymbol{A}^{mathbf{2}} )
[
mathbf{7 A}+mathbf{1 0}=mathbf{0}
]
12
855 Is it possible to define the matrix ( A+B )
when
A has 3 rows and ( mathrm{B} ) has 2 columns
12
856 ( mathbf{f}left[begin{array}{ccc}boldsymbol{x}-boldsymbol{y} & mathbf{1} & boldsymbol{z} \ mathbf{2} boldsymbol{x}-boldsymbol{y} & boldsymbol{0} & boldsymbol{w}end{array}right]=left[begin{array}{ccc}-mathbf{1} & mathbf{1} & mathbf{4} \ mathbf{0} & mathbf{0} & mathbf{5}end{array}right] ) find
( boldsymbol{x}, boldsymbol{y}, boldsymbol{z}, boldsymbol{w} )
12
857 Solve the following system of linear equations using matrix method: ( 3 x+ )
( boldsymbol{y}+boldsymbol{z}=mathbf{1}, mathbf{2} boldsymbol{x}+mathbf{2} boldsymbol{z}=mathbf{0}, mathbf{5} boldsymbol{x}+mathbf{5} boldsymbol{y}+ )
( mathbf{2} z=mathbf{2} )
12
858 If ( boldsymbol{A}=left(boldsymbol{a}_{boldsymbol{i} j}right)_{mathbf{2} times mathbf{2}}, ) where ( boldsymbol{a}_{boldsymbol{i} j}=boldsymbol{i}+boldsymbol{j}, ) then
( A ) is equal to:
A. ( left[begin{array}{ll}1 & 2 \ 2 & 3end{array}right] )
B. ( left[begin{array}{ll}0 & 1 \ 1 & 0end{array}right] )
( mathbf{c} cdotleft[begin{array}{ll}2 & 3 \ 3 & 4end{array}right] )
D. ( left[begin{array}{ll}1 & 2 \ 3 & 4end{array}right] )
12
859 A square, non-singular matrix ( boldsymbol{A} )
satifies ( boldsymbol{A}^{2}-boldsymbol{A}+mathbf{2} boldsymbol{I}=mathbf{0}, ) then ( boldsymbol{A}^{-1}= )
( mathbf{A} cdot I-A )
в. ( frac{(I-A)}{2} )
c. ( I+A )
D. ( frac{(I+A)}{2} )
12
860 If ( boldsymbol{A}+boldsymbol{B}=left[begin{array}{ll}mathbf{2} & mathbf{3} \ mathbf{4} & mathbf{5}end{array}right] ) and ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2} \ mathbf{0} & mathbf{3}end{array}right] )
then matrix ( B ) is
( mathbf{A} cdotleft[begin{array}{ll}1 & 1 \ 4 & 2end{array}right] )
в. ( left[begin{array}{ll}1 & 4 \ 1 & 2end{array}right] )
c. ( left[begin{array}{ll}2 & 4 \ 1 & 1end{array}right] )
D. ( left[begin{array}{ll}4 & 2 \ 1 & 1end{array}right] )
12
861 If ( mathbf{A}=left[begin{array}{ll}mathbf{3} & -mathbf{4} \ mathbf{1} & -mathbf{1}end{array}right] ) then ( boldsymbol{A}^{k}= )
( left[begin{array}{cc}mathbf{1}+mathbf{2} boldsymbol{k} & -mathbf{4} boldsymbol{k} \ boldsymbol{k} & mathbf{1}-mathbf{2} boldsymbol{k}end{array}right] )
where ( k ) is any ( + ) ve integer
12
862 If ( A ) is a square matrix of order 3
then ( left|mathbf{A} mathbf{d} mathbf{j}left(A d j A^{2}right)right|= )
( mathbf{A} cdot|A|^{2} )
B . ( |A|^{4} )
c. ( left.|A|^{8}right|^{8} mid )
D. ( |A|^{16} )
12
863 If ( boldsymbol{A}=left[begin{array}{cc}cos 2 boldsymbol{theta} & -sin 2 boldsymbol{theta} \ sin 2 boldsymbol{theta} & cos 2 boldsymbol{theta}end{array}right] ) and ( boldsymbol{A}+ )
( A^{T}=I, ) where ( I ) is the unit of matrix of
( 2 times 2 ) and ( A^{T} ) is the transpose of ( A, ) then
the value of ( theta ) is equal to
A ( cdot frac{pi}{6} )
в.
c. ( pi )
D. ( frac{3 pi}{2} )
12
864 If ( A ) is a skew-symmetric matrix, then trace of ( A ) is
( mathbf{A} cdot mathbf{1} )
B. – –
( c cdot 0 )
D. none of these
12
865 If ( boldsymbol{A}=left[begin{array}{ll}mathbf{1} & mathbf{2}end{array}right], boldsymbol{B}=left[begin{array}{ll}mathbf{3} & mathbf{4}end{array}right] ) then ( boldsymbol{A}- )
( boldsymbol{B}= )
( mathbf{A} cdotleft[begin{array}{ll}-2 & -2end{array}right] )
B. ( left[begin{array}{ll}2 & 2end{array}right] )
( mathbf{c} cdotleft[begin{array}{ll}-3 & -1end{array}right] )
D. None of these
12
866 Let ( A ) be a ( 2 times 2 ) matrix with non-zero
entries and let ( mathbf{A}^{2}=boldsymbol{I}, ) where I is ( mathbf{2} times mathbf{2} )
identity matrix. Define ( operatorname{Tr}(mathbf{A})=operatorname{sum} ) of diagonal elements of ( A ) and ( |mathbf{A}|= ) determinant of matrix A.

Statement-1 ( operatorname{Tr}(mathrm{A})=0 ) Statement-2:
( |mathbf{A}|=mathbf{1} )
A. Statement-1 is true, Statement-2 is true; Statement- is not the correct explanation for statement-
B. Statement-1 is true, Statement-2 is false
c. Statement- 1 is false, statement- 2 is true
D. Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-

12
867 f ( boldsymbol{A}=left[begin{array}{ll}boldsymbol{x} & mathbf{1} \ mathbf{1} & mathbf{0}end{array}right], boldsymbol{A}^{2}=boldsymbol{I} ) then find ‘ ( boldsymbol{x}^{prime} ) 12
868 Construct a ( 3 times 2 ) matrix whose
elements are given by ( a_{i j}=2 i-j )
12
869 If ( A timesleft(begin{array}{l}1 \ 0end{array}right)=(12), ) then the order of ( A ) is
A ( .2 times 1 )
B. ( 2 times 2 )
c. ( 1 times 2 )
D. 3 ( times 2 )
12
870 if ( A=left[begin{array}{cc}2 & 3 \ 5 & -7end{array}right] ) then ( quadleft(A^{1}right)^{2}= )
A. ( left[begin{array}{ccc}5 & -7 & 12 \ 1 & 4 & 22end{array}right] )
B ( cdotleft[begin{array}{cc}1 & 17 \ 1 & -4 \ 0 & 2end{array}right] )
c. ( left[begin{array}{cc}-19 & -25 \ -15 & 64end{array}right] )
D. ( left[begin{array}{cc}19 & -25 \ -15 & 64end{array}right] )
12
871 If ( A, B ) are symmetric matrices of the same order then ( mathbf{A B}-mathbf{B A} ) is
A. symmetric matrix
B. skew symmetric matrix
c. Diagonal matrix
D. identity matrix
12
872 ( mathbf{f} boldsymbol{A}=left[begin{array}{lll}mathbf{1} & mathbf{4} & mathbf{0} \ mathbf{2} & mathbf{5} & mathbf{0} \ mathbf{3} & mathbf{6} & mathbf{0}end{array}right] ) and ( boldsymbol{B}=left[begin{array}{lll}mathbf{3} & mathbf{2} & mathbf{1} \ mathbf{1} & mathbf{2} & mathbf{3} \ mathbf{4} & mathbf{5} & mathbf{6}end{array}right] )
and ( C=left[begin{array}{lll}3 & 2 & 1 \ 1 & 2 & 3 \ 7 & 8 & 9end{array}right], ) Then evaluate
( operatorname{matrix} boldsymbol{A B}-boldsymbol{B C} )
12
873 If ( A ) is a ( 3 times 3 ) skew-symmetric matrix,
then trace of ( A ) is equal to
( A cdot 1 )
в. ( |A| )
( c cdot-1 )
D. none of these
12
874 ( mathbf{A}=left[begin{array}{cc}cos alpha & sin alpha \ -sin alpha & cos alphaend{array}right] ) then ( mathbf{A} . mathbf{A}^{mathbf{T}} )
A. Null matrix
в. А
( c cdot I )
D. A
12
875 If ( A ) and ( B ) are symmetric matrices,
then write the condition for which ( A B ) is
also symmetric
12
876 Define a scalar matrix. 12
877 If square matrices ( A ) and ( B ) are such
that ( boldsymbol{A} boldsymbol{A}^{prime}=boldsymbol{A}^{prime} boldsymbol{A}, boldsymbol{B} boldsymbol{B}^{prime}=boldsymbol{B}^{prime} boldsymbol{B}, boldsymbol{A} boldsymbol{B}^{prime}= )
( boldsymbol{B}^{prime} boldsymbol{A} )
then is the statement ( A B(A B)^{prime}= )
( (A B)^{prime} A B ) is
where ( A^{prime} ) is transpose of ( A )
If true enter 1 else enter 0
12
878 [
text { If } boldsymbol{f}(boldsymbol{x}, boldsymbol{y})=boldsymbol{x}^{2}+boldsymbol{y}^{2}-boldsymbol{2} boldsymbol{x} boldsymbol{y},(boldsymbol{x}, boldsymbol{y} in boldsymbol{R})
]
and
( boldsymbol{A}= )
[
left[begin{array}{lll}
fleft(x_{1}, y_{1}right) & fleft(x_{1}, y_{2}right) & fleft(x_{1}, y_{3}right) \
fleft(x_{2}, y_{1}right) & fleft(x_{2}, y_{2}right) & fleft(x_{2}, y_{3}right) \
fleft(x_{3}, y_{1}right) & fleft(x_{3}, y_{2}right) & fleft(x_{3}, y_{3}right)
end{array}right]
]
such that trace ( (boldsymbol{A})=mathbf{0}, ) then which of the following is true (only one option)
A. ( operatorname{det}(A) geq 0 )
в. ( operatorname{det}(A)=0 )
( mathbf{c} cdot operatorname{det}(A) leq 0 )
D. ( operatorname{det}(A)>0 )
12
879 Two ( n times n ) square matrices ( A ) and ( B ) are
said to be similar if there exists a non-
singular matrix ( boldsymbol{P} ) such that
( boldsymbol{P}^{-1} boldsymbol{A} boldsymbol{P}=boldsymbol{B} )
If ( A ) and ( B ) are two non-singular
matrices, then
( mathbf{A} cdot A ) is similar to ( B )
B. ( A B ) is similar to ( B A )
C. ( A B ) is similarto ( A^{-1} B )
D. none of these
12
880 If ( boldsymbol{A}-boldsymbol{A}^{prime}=mathbf{0}, ) then ( boldsymbol{A}^{prime} ) is
A. orthogonal matrix
B. symmetric matrix
c. skew-symmetric matrix
D. triangular matrix
12
881 If ( boldsymbol{A}=left[begin{array}{cc}cos boldsymbol{alpha} & -sin boldsymbol{alpha} \ sin boldsymbol{alpha} & cos boldsymbol{alpha}end{array}right], ) then ( boldsymbol{A}+ )
( A^{prime}=I, ) if the value of ( alpha ) is
A ( cdot frac{pi}{6} )
в.
( c )
D. ( frac{3 pi}{2} )
12
882 If ( boldsymbol{A}=left[begin{array}{cc}1 & -1 \ -1 & 1end{array}right], ) satisfies the matrix
equation ( A^{2}=k A, ) write the value of ( k )
12

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