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.

Matrices Questions

List of matrices Questions

Question NoQuestionsClass
1( fleft(begin{array}{ccc}-1 & 2 & 4 \ 3 & 6 & -5end{array}right] ) then find ( 3 A )12
2The 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
3Trace 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
4If ( 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
5If ( 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
6Q 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
7Assertion ( 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
8Suppose ( 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
10Find 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
11Find 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
12If ( 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
13Let ( 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
14Which 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
15A matrix has 8 elements. What are the
possible orders it can have?
12
16If ( 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
17If ( A ) is a skew-symmetric matrix, then
trace of ( boldsymbol{A} ) is
12
18If ( A=left[begin{array}{cr}2 & 5 a \ -3 & 1end{array}right] ) and ( A ) doesn’t have multiplicative inverse then find ( mathbf{A} )12
19For 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
20matrix 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
22If ( 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
23Let ( 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
24cos
-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
25Find 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
26If ( 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
27For 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
29In 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
30Assertion
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
31If ( 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
32If ( 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
33If ( 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
34Find 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
35If ( 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
36If 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
37If ( 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
38Consider ( 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
39Find the adjoint of the matrix ( A= ) ( left[begin{array}{lll}1 & 4 & 3 \ 4 & 2 & 1 \ 3 & 2 & 2end{array}right] )12
40If ( 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
41If 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
42Using 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
43Assertion
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
44If ( 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
45Find ( 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
46The 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
47Assertion
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
48If ( 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
49Find 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
50If ( 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
52If ( boldsymbol{A}=left(begin{array}{c}-121 \ 1 & 23end{array}right), B=(1) ) and ( C=(21) )
verify ( (A B) C=A(B C) )
12
53Matrix 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
54Out 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
55If ( 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
56Write 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
58If ( mathbf{A} ) is a non-singular square matrix of order ( 3 times 3, ) find ( |a d j A| )12
59fthe 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
60Three roots of ( n ) are
A. 0,1,2
в. -1,1,3
( mathrm{c} .-2,2,3 )
D. -3,1,5
12
61Find 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
64If ( 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
70A 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
71If ( 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
72Identify the order of matrix ( left[begin{array}{ccc}1 & 0 & -4 \ 2 & -1 & 3end{array}right] )12
73If ( 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
74State 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
75If ( 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
76Find 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
77Number 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
78ff ( C=left[begin{array}{cc}3 & -6 \ 0 & 9end{array}right] ) find
¡) 2C
ii) ( frac{1}{3} C )
iii) –
12
79Find the inverse of matrices by
elementary row transformation.
12
80Let ( 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
81If ( 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
82Find 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
87If ( 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
88ff ( 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
89Elements 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
91If ( 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
92Using elementary row transformation, find the inverse of ( left[begin{array}{ccc}2 & 0 & -1 \ 5 & 1 & 0 \ 0 & 1 & 3end{array}right] )12
93If ( 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
94ff ( 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
95Let ( 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
96If ( 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
97B 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
98If ( boldsymbol{A}=left(begin{array}{l}23 \ 4 \ 5end{array}right), ) then find the transpose of ( boldsymbol{A} )12
99If ( [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
100Evaluate
( 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
101If ( 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
103If ( 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
105Consider 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
107If ( 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
108If ( 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
110If ( 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
111If ( 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
113If ( 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
114If
(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
115If ( 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
116Using 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
117For ( 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
118Prove that ( (A B)(A B)^{-1}=1 )12
119If 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
120Find 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
121If ( A ) and ( B ) are symmetric matrices, then ( (A B-B A) ) is skew-symmetric.
A. True
B. False
12
122If ( 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
123If ( 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
124If ( 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
125If ( boldsymbol{A}=left[begin{array}{c}-2 \ 4 \ 6end{array}right] ) and ( B=[14-6] ) then
find ( boldsymbol{A B} )
12
126Find the inverse of the matrix
( left[begin{array}{ccc}1 & 0 & 0 \ 3 & 3 & 0 \ 5 & 2 & 1end{array}right] )
12
127If ( 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
12921.
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
130If ( 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
131The 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
132Find ( x, ) if the matrix ( left|begin{array}{ccc}-1 & 2 & 3 \ 2 & 5 & 6 \ 3 & x & 7end{array}right| ) is a
symmetric matrix.
12
13312. 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
134Construct 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
135Find the order of the matrix
( left[begin{array}{ccc}1 & 1 & 3 \ 5 & 2 & 6 \ -2 & -1 & -3end{array}right] )
12
136What 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
137Let ( 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
139Find ( 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
140If 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
142Using 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
143Suppose ( 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
144If ( 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
145If ( 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
146Suppose ( 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
147If ( 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
149f 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
150Let ( 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
151Inverse 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
152If ( 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
153If ( 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
154If ( alpha ) and ( beta ) differ by an odd multiply on ( pi / 2 ) prove that the product of the two matrices given below is a null matrix12
155If ( boldsymbol{A}=left[begin{array}{l}mathbf{1} \ mathbf{2} \ mathbf{3}end{array}right], ) then find ( boldsymbol{A} boldsymbol{A}^{boldsymbol{T}} )12
156If ( 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
158Choose 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
159If ( 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
160Which 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
161If ( 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
162Write a ( 2 times 2 ) matrix which is both
symmetric and skew-symmetric.
12
1637.
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
164If ( 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
165If ( 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
166If ( 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
168ff ( 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
169If ( 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
170Find 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
171If ( 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
172If ( 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
176Assertion
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
177If ( 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
178Find the rank of the matrix ( boldsymbol{A}= )
( left[begin{array}{ccc}1 & 5 & 9 \ 4 & 8 & 12 \ 7 & 11 & 15end{array}right] )
12
179A 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
180If ( 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
181If ( 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
182Construct a ( 2 times 3 ) matrix ( A=left[a_{i j}right] )
whose element ( a_{i j} ) is ( a_{i j}=2 i-j )
12
183If ( 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
184Multiply 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
186If ( 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
188Solve 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
189If ( 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
190To find the inverse of the matrix ( boldsymbol{A}= )
( [121 ; 011 ; 311] ) by elementary
transformation method
12
191ff ( 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
192A matrix consists of 30 elements. What
are the possible orders it can have?
12
193ff ( 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
194Let ( 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
196If ( 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
197If ( 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
198f 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
199toppr 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
200Assertion
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
201If ( 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
202IF ( A ) be a square matrix, then ( boldsymbol{A}+boldsymbol{A}^{boldsymbol{T}} )
is a symmetric matrix?
A. True
B. False
12
203If ( 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
204Let ( 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
206If ( 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
208If ( 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
209For 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
211Express 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
212If 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
213If ( 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
215Three 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
217Prove that the matrix ( B^{prime} A B ) is
symmetric or skew symmetric
according as ( A ) is symmetric or skew symmetric.
12
218If ( 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
219If 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
221Determine of ( U ) is
A . 13
B. 15
( c .3 )
D. 2
12
222( |f| A mid=47, ) then find ( left|A^{T}right| )12
223Express 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
224The transpose of a rectangular matrix is
( mathbf{a} )
A. rectangular matrix
B. diagonal matrix
c. square matrix
D. scalar matrix
12
225Which 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
227Let ( 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
228Find 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
230Construct 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
231If ( 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
233The 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
234Find 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
235If ( A=[123] ) and ( B=left[begin{array}{l}1 \ 2 \ 3end{array}right], ) then find
( (A B)^{prime} )
12
236Let ( 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
237Assertion
( 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
238For 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
239If ( 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
240Find 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
241If ( 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
242If ( 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
243Let ( 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
244If ( 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
245If ( 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
247f ( 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
248If ( 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
249In a skew-symmetric matrix, the diagonal elements are all
A. one
B. zero
c. different from each other
D. non-zero
12
250ff ( 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
252If ( 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
253Find 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
254Match 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
256Express 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
258If ( 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
259If ( 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
260If ( 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
262In 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
263A 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
264If ( 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
266If ( 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
267If ( 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
268If 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
269The 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
270If ( 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
271Assertion ( 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
272If ( 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
273If ( 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
274If 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
275The transpose of a column matrix is
A. zero matrix
B. diagonal matrix
c. column matrix
D. row matrix
12
276Using elementary transformations find
the inverse of
( left[begin{array}{ccc}3 & 2 & 1 \ 2 & 4 & 3 \ 2 & -1 & 2end{array}right] )
12
277If ( 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
278If ( 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
279If ( 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
280Write ( 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
283Using 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
285Express 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
287If ( 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
288Suppose ( 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
290If ( 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
291If ( 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
292Find ( 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
293If ( 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
294Construct a ( 2 times 3 ) matrix ( A=left[a_{i j}right] )
whose elements are given by ( a_{i j}= )
( |2 i-3 j| )
12
295If ( 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
296Find 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
297The 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
298If ( 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
299Construct 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
301A 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
302Solve 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
303Assertion
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
304Choose 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
306If ( 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
307If ( 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
309Let ( 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
311Multiplication 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
312Let ( 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
313Let ( 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
314Let ( 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
315Find 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
316If ( 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
318If ( 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
319If 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
321If ( 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
323If 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
324Let ( 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
325Is it possible to define the matrix AB and BA when :
A has 4 rows and ( mathrm{B} ) has 4 columns
12
326Using 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
327In 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
328Matrices ( 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
329If ( 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
330If ( 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
331Assertion ( 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
332By 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
333If ( 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
334By using elementary transformation
find the inverse of the matrix ( boldsymbol{A}= ) ( left[begin{array}{ll}1 & 2 \ 2 & 1end{array}right] )
12
335If ( [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
336The 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
337Find 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
338Solve 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
339Solve 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
340If 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
342Using 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
343If ( 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
344The 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
345If 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
346If ( 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
347Prove 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
348If ( 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
349Convert ( [1 quad-12 )
3] into an identity
matrix by suitable row transformations.
12
350If ( 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
351If ( 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
352If ( 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
353For 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
354If ( 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
355type 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
356Construct 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
358Find ( 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
360Prove 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
361By 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
362For 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
363If ( 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
364If ( 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
366If ( 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
367If ( 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
369If ( 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
372Find 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
374If ( 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
375If ( 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
376Matrices ( 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
378If ( 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
379Find 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
380Let ( 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
381Assertion
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
382If 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
38411.
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
385Find 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
386For 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
387State 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
388Find 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
389Let ( 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
390What 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
391The 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
394Let ( 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
395Find ( : 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
396If ( 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
397IF ( 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
398If ( 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
399If ( 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
400Write a ( 2 times 2 ) matrix which is both
symmetric and skew symmetric.
12
401If ( 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
402Matrices obtained by changing rows and columns is called
A. rectangular matrix
B. transpose
c. symetric
D. None of the Above
12
403A 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
404Without 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
405Let ( 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
406ff ( 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
407f ( 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
410Solve: ( 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
411If ( 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
412Let ( 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
413Find inverse of the matrix
( left[begin{array}{ccc}1 & -1 & 1 \ 0 & 1 & 1 \ 3 & 2 & -4end{array}right] ) by elementary
transformation.
12
414If ( 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
415For ( 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
416The 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
417Construct 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
418If ( 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
419If ( 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
420If ( 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
421Assertion ( 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
422Let ( 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
423For 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
424If ( 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
426If 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
42710. 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
428A 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
429If ( 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
430f ( 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
431Simplify:
( 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
432Compute ( 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
433If ( 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
434ff ( boldsymbol{A}=left[begin{array}{cc}mathbf{2} & mathbf{- 1} \ mathbf{- 1} & mathbf{2}end{array}right] ) Find ( boldsymbol{A}^{2} )12
435If ( 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
436If ( 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
437Find 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
438If ( 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
439Identify 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
441Identify 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
442covert 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
443If ( 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
444If ( 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
445If 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
446Find ( 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 ) then12
448fthe 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
449Compute 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
451ff ( 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
452fthe 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
453If ( 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
455If ( 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
456Trace of ( A^{50} ) equals
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
12
457Define a scalar matrix.12
458Find 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
459A 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
460If ( 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
461If ( 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
462If ( 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
464Find ( 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
465ff ( 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
466If ( 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
467Construct 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
469By 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
471If ( 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
472Construct 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
473Find 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
474If ( 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
475ff ( 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
4773.
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
478If ( 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
479What is meant by transposing of a
matrix? Give an example.
12
480Solve 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
481If ( 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
482ff ( 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
484If ( 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
485If ( 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
487Using 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
488Let ( 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
489If ( 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
490If ( 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
491not 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
492If ( 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
493Convert ( left[begin{array}{cc}1 & -1 \ 2 & 3end{array}right] ) into on identity matrix by suitable raw transformation.12
494Let ( 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
495Matrix ( 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
496Using elementary transformation, find the inverse of the matrix ( left[begin{array}{ccc}2 & -3 & 3 \ 2 & 2 & 3 \ 3 & -2 & 2end{array}right] )12
497Let ( 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
498If ( 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
499The 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
501Inverse of a diagonal non-singular matrix is
A. Scalar matrix
B. Skew symmetric matrix
c. zero matrix
D. Diagonal matrix
12
502If ( 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
503Construct 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
505If ( 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
506Find the inverse of the following matrix by using elementary row transformation
( left[begin{array}{ll}2 & 5 \ 1 & 3end{array}right] )
12
507ff ( 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
50825. 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
509If ( 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
510If ( 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
512ff ( 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
514If ( 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
515If ( 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
517If matrix ( boldsymbol{A}=[mathbf{1} mathbf{2} mathbf{3}], ) then find ( boldsymbol{A} boldsymbol{A}^{boldsymbol{T}} )12
518Given ( 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
519If ( 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
521If ( A=left(begin{array}{ccc}1 & -1 & 3 \ 5 & -4 & 7 \ 6 & 0 & 9 & 8end{array}right), ) Find the order of the
matrix
12
522Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] )12
523If
( 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
524Construct 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
526Select 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
527Prove 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
528If ( 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
529f ( 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
531fthe 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
532Find ( 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
533If ( 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
534If ( 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
535If ( 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
538If 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
539Evaluate
( left[begin{array}{l}3 \ 4 \ 1end{array}right]left[begin{array}{lll}2 & -1 & 3end{array}right] )
12
540The inverse of a symmetric matrix is
A. symmetric
B. skew-symmetric
c. diagonal matrix
D. singular matrix
12
541Show 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
543Identify a matrix
В. ( A={1,2} )
( mathbf{c} cdot A=left[begin{array}{ll}1 & 2end{array}right] )
D. None of these
12
544Let ( 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
545Ti 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
546If ( 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
547If ( A ) is a square matrix, then ( A-A^{T} ) is
A. unit matrix
B. null matrix
( c . A )
D. a skew symmetric matrix
12
548If ( 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
549If ( 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
550If ( 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
551f ( 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
552If ( 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
553Given, 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
554Find 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
555For ( 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
557The 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
559Assertion ( 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
561Identify 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
562A 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
563Let 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
565If 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
566If ( 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
569Find 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
570If ( A B=A ) and ( B A=B ) then ( B^{2} ) is
equal to
A. ( B )
в. ( A )
( c .-B )
D. ( B^{2} )
12
571Which 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
572The 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
574Unit 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
575The 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
576If ( 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
577The 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
579Let ( 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
580If ( 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
581For any square matrix ( boldsymbol{A} ) with real numbers,
( A+A^{prime} ) is a symmetric and
( A-A^{prime} ) is a skew-symmetric
12
58216. 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
583sin -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
586Find 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
588If ( 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
589If ( 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
590If ( 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
592A 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
593Let 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
594If ( 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
595The 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
596Construct 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
597If ( boldsymbol{A}=left[begin{array}{ll}2 & 3 \ 5 & 7end{array}right], ) then find ( A+A^{T} )12
598ff ( 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
59918.
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
600If ( 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
601f ( 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
602If ( 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
603If ( 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
604If ( 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
605If ( 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
606If ( 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
607The 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
609Which 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
610If ( A ) is a skew-symmetric matrix of
order ( 3, ) then prove that det ( A=0 )
12
611Show that all the diagonal elements of a skew-symmetric matrix are zero.12
612If ( 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
613Find ( 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
614ff ( 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
615For 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
616If ( 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
617fthe 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
618Simplify ( : 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
619Given 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
620If ( 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
621If ( 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
622Find the inverse of the following matrix
using transformation method. ( left[begin{array}{cc}2 & -3 \ -1 & 2end{array}right] )
12
623By 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
624If 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
625If 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
626ff ( 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
627Let ( 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
629Find 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
631If ( 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
632If ( 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
633If ( 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
634Q 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
635Let ( 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
636If ( 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
637Two ( 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
638Two ( 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
639If ( 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
640Matrices ( 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
641What 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
642If ( 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
643Express the square matrix ( A ) as the
sum of a symmetric and a skewsymmetric matrix.
12
644Find 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
645Using 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
646If ( 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
647There 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
648f ( 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
649If ( 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
650Find 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
651The 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
652Find the inverse of the matrix
( left[begin{array}{ccc}1 & 0 & 0 \ 3 & 3 & 0 \ 5 & 2 & 1end{array}right] )
12
65342. 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
654Let ( 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
655If ( 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
656If 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
657If ( 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
658If ( 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
659Consider 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
660If ( 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
661If ( 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
662The number of different possible orders of matrices having 18 identical elements is
( A cdot 3 )
B.
( c cdot 6 )
D. 4
12
663If ( A ) is a square matrix with ( |A|=8 )
Find the value of ( left|boldsymbol{A} boldsymbol{A}^{-1}right| )
12
664A 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
666If ( 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
667The 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
669If ( 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
670If ( 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
671ff ( 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
672Assertion ( 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
673The 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
674Construct 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
676If ( 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
677Let ( 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
678If 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
679Find the inverse of the following matrices by the adjoining method ( left[begin{array}{cc}-1 & 5 \ -3 & 2end{array}right] )12
680Construct 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
681Two 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
6845.
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
685If ( 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
686If ( 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
687A 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
688a
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
689If ( 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
690Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] )
whose elements are given by ( a_{i j}=2 i- )
( boldsymbol{j} )
12
691Find 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
692If
( 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
693If ( 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
694Construct a ( 2 times 2 ) matrix ( A=left[a_{i j}right] )12
695If ( 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
696If ( 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
699f ( 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
700Define a diagonal matrix.12
701Let ( 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
702ff ( 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
703If ( 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
704If ( 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
705Find the transpose of matrix ( left[begin{array}{ll}2 & 5 \ 1 & 3end{array}right] )12
706Determine 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
707If ( 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
708Let ( 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
711The 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
712The 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
713If 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
714If ( 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
715If ( boldsymbol{A}=left[begin{array}{ll}mathbf{0} & mathbf{0} \ mathbf{4} & mathbf{0}end{array}right], ) then find ( boldsymbol{A}^{16} )12
716Find 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
718If ( 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
719If ( 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
720Construct 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
721Find 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
723Using 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
724Find 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
725If ( 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
727If ( 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
729If ( A ) and ( B ) are two square matrix of
order ( n ) then prove that ( :(A B)^{-1}= )
( boldsymbol{B}^{-1} boldsymbol{A}^{-1} )
12
730If ( 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
731If ( 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
733Show 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
735If ( 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
736Find Order of matrix : ( left[begin{array}{lll}mathbf{1} & mathbf{2} & mathbf{6} \ mathbf{2} & mathbf{4} & mathbf{3}end{array}right] )12
737If ( 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
739Let 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
740If ( 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
741ff ( 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
742If ( 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
743Find 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
744If ( 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
745If ( 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
746If ( 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
747Construct 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
750If ( 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
753Given 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
754The matrix ( boldsymbol{B} ) is
A. Symmetric
B. Scalar
c. Skew hermitian
D. Skew- symmetric
12
755Find 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
756If ( 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
757If ( 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
759If 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
760Find 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
761If ( 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
762If a ( M ) matrix ( A ) is such that ( A A^{T}= )
( boldsymbol{I}=boldsymbol{A}^{boldsymbol{T}} boldsymbol{A}, ) find ( |boldsymbol{A}|=? )
12
763If 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
765If ( 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
766If ( 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
767If ( 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
768Using 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
769Is 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
770If ( 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
772The 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
773Find the inverse of the following matrix by using elementary row transformation:
( left[begin{array}{ll}5 & 2 \ 2 & 1end{array}right] )
12
774For 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
775If ( 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
776If ( 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
777If ( 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
778The 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
779If ( 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
780Find the inverse of ( left[begin{array}{ccc}3 & -1 & -2 \ 2 & 0 & -1 \ 3 & -5 & 0end{array}right] )
using elementary row transformations.
12
781If ( 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
782The 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
783If ( 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
784If ( 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
785Let ( 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
786If ( 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
787If 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
788If 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
789If ( 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
790If ( 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
791If ( 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
792Solve 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
793The number of nonzero diagonal
matrices of order 3 satisfying ( A^{2}=A ) is
12
794Using 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
795If ( 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
796If ( 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
797The transpose of a row matrixis
A. zero matrix
B. diagonal matrix
C. column matrix
D. row matrix
12
798Write 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
799Find 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
800If ( 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
801If ( 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
802Find ( 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
803For 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
804atrices 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
805If ( 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
806Find 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
808Assertion
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
809If ( 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
810Let ( A ) and ( B ) be square matrices of the
other ( 3 times 3 . ) Is ( (A B)^{2}=A^{2} B^{2} ? ) Give
reasons.
12
811ff ( 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
812If ( 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
814Inverse 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
815The 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
816If ( 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
818Let ( 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
819Find 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
820IIf ( 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
824Solve 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
825Show that square matrix ( A ) and its
transpose ( A^{T} ) have the same eigen
values.
12
826If ( 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
827Find 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
828Assertion 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
829Write 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
831If ( 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
832Two 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
834Find ( 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
835Find 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
837ff ( 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
838If ( 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
839If ( 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
840If ( 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
841Let ( A ) be a ( 2 times 2 ) matrix and ( B=A+ )
( A^{T} . ) Then show that ( B ) is a symmetric
matrix.
12
842If ( 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
843Let ( A ) being a square matrix, then prove that ( boldsymbol{A}+boldsymbol{A}^{T} ) is symmetric.12
844If ( 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
845Determine 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
846Find 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
847If ( 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
848If ( 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
849If ( [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
850Find ( 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
851If ( 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
853If ( 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
855Is 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
857Solve 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
858If ( 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
859A 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
860If ( 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
861If ( 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
862If ( 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
863If ( 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
864If ( 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
865If ( 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
866Let ( 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
867f ( 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
868Construct a ( 3 times 2 ) matrix whose
elements are given by ( a_{i j}=2 i-j )
12
869If ( 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
870if ( 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
871If ( 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
873If ( 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
875If ( A ) and ( B ) are symmetric matrices,
then write the condition for which ( A B ) is
also symmetric
12
876Define a scalar matrix.12
877If 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
879Two ( 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
880If ( 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
881If ( 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
882If ( 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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