Q: If matrix A= [ array ccc 3 & -2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1 array ] and A^ -1 = 1 k adj (A), then k is

Given, A= [ array ccc 3 & -2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1 array ] and A^ -1 = 1 k adj (A)....(i) We know that, A^ -1 = adj (A) |A| ...(ii) By comparing Eqs. (i) and (iii), k=|A| So, aligned |A| & = | array ccc 3 & -2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1 array | \\ & =3(2+1)+2(1+0)+4(1-0)=15 aligned

Q: If A= [ array ccc 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 array ] and B= [ array ccc a^2 & a b & a c \\ a b & b^2 & b c \\ a c & b c & c^2 array ] , then A B is equal to

aligned & A B= [ array ccc 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 array ] [ array ccc a^2 & a b & a c \\ a b & b^2 & b c \\ a c & b c & c^2 array ] \\ & A B= [ array ccc a b c-a b c & b^2 c-b^2 c & b c^2-b c^2 \\ -a^2 c+a^2 c & -a b c+a b c & -a c+a c \\ a^2 b-a^2 b & a b^2-a b^2 & a b c-a b c array ] \\ & = [ array lll 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 array ]= O \\ & aligned

Q: If A = [ array cc 0 & 2 \\ 3 & -4 array ] and k ~A = [ array cc 0 & 3 a \\ 2 b & 24 array ], then the values of k, a and b respectively are:

k A= [ array cc 0 & 2 k \\ 3 k & -4 k array ]= [ array cc 0 & 3 a \\ 2 b & 24 array ] k=-6, a=-4 and b=-9

Q: Let A= [ array ccc 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 array ] and 10 B= [ array ccc 4 & 2 & 2 \\ -5 & 0 & \\ 1 & -2 & 3 array ] If B is the inverse of A, then the value of is

A= [ array ccc 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 array ] |A|= [ array ccc 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 array ] =1(1+3)+1(2+3)+1(2-1)=4+5+1=10 Adj A= [ array ccc 4 & -5 & 1 \\ 2 & 0 & -2 \\ 2 & 5 & 3 array ]^T= [ array ccc 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 array ] B=A^ -1 = 1 10 [ array ccc 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 array ] [ A^ -1 = 1 |A| Adj A ] 10 B= [ array ccc 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 array ] Hence, =5

Q: If [ array cc 1 & - \\ & 1 array ] [ array cc 1 & \\ - & 1 array ]^ -1 = [ array cc a & -b \\ b & a array ], then

Given, [ array cc 1 & - \\ & 1 array ] [ array cc 1 & \\ - & 1 array ]^ -1 = [ array cc a & -b \\ b & a array ] [ array cc 1 & - \\ & 1 array ] 1 1+ ^2 [ array cc 1 & - \\ & 1 array ] = [ array cc a & -b \\ b & a array ] 1 1+ ^2 [ array cc 1- ^2 & -2 \\ 2 & 1- ^2 array ]= [ array cc a & -b \\ b & a array ] [ array cc 1- ^2 1+ ^2 & 2 1+ ^2 \\ 2 1+ ^2 & 1- ^2 1+ ^2 array ]= [ array cc a & -b \\ b & a array ] [ array cc 2 & - 2 \\ 2 & 2 array ]= [ array cc a & -b \\ b & a array ] a= 2 , b= 2

Question 1

If A= 1 3 [ array ccc 1 & 2 & 2 \ 2 & 1 & -2 \ a & 2 & b array ] is an orthogonal matrix, then

Accepted Answer

We know that, if A is an orthogonal matrix, then gathered \ AA^T=I \ \ [ array ccc 1 & 2 & 2 \ 2 & 1 & -2 \ a & 2 & b array ] 1 3 [ array ccc 1 & 2 & a \ 2 & 1 & 2 \ 2 & -2 & b array ]= [ array lll 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 array ] \ [ array ccc 1 & 2 & 2 \ 2 & 1 & -2 \ a & 2 & b array ] [ array ccc 1 & 2 & a \ 2 & 1 & 2 \ 2 & -2 & b array ]= [ array lll 9 & 0 & 0 \ 0 & 9 & 0 \ 0 & 0 & 9 array ] \ [ array ccc 9 & 0 & a+4+2 b \ 0+4+2 b & 9 & 2 a+2-2 b \ 2 a+2-2 b & a^2+4

Question 2

If matrix A= [ array ccc 3 & -2 & 4 \ 1 & 2 & -1 \ 0 & 1 & 1 array ] and A^ -1 = 1 k adj (A), then k is

Accepted Answer

Given, A= [ array ccc 3 & -2 & 4 \ 1 & 2 & -1 \ 0 & 1 & 1 array ] and A^ -1 = 1 k adj (A)....(i) We know that, A^ -1 = adj (A) |A| ...(ii) By comparing Eqs. (i) and (iii), k=|A| So, aligned |A| & = | array ccc 3 & -2 & 4 \ 1 & 2 & -1 \ 0 & 1 & 1 array | \ & =3(2+1)+2(1+0)+4(1-0)=15 aligned

Question 3

If A and B are symmetric matrices of same order such that A B+B A=X and A B-B A=Y, then (X Y)^ T =

Accepted Answer

Since X Y=(A B+B A)(A B-B A) =( AB ) AB +( BA )( AB )-( AB )( BA )-( BA )( BA ) Now (X Y)^ T =(( AB ) ( AB ))^ T +( BA AB )^ T - ( AB BA )^ T array r =( AB )^ T ( AB )^ T +( AB )^ T ( BA )^ T -( BA )^ T ( AB )^ T \ -( BA )^ T ( BA )^ T array = (B^ A^ ) (B^ A^ )+ (B^ A^ ) (A^ B^ ) - ( A ^ B ^ ) ( B ^ A ^ )- ( A ^ B ^ ) ( A ^ B ^ ) Since, A & B are symmetric matrix. =( BA )( BA )+( BA )( AB )-( AB )( BA ) -( AB )( AB ) =( BA - AB )( BA + AB )=- YX

Question 4

If A= [ array cc 1 & 0 \ 0 & -1 array ], P= [ array ll 1 & 1 \ 0 & 1 array ] and X=A P A^T, then A^T X^ 50 A=

Accepted Answer

Since A A^T=I therefore matrix A is orthogonal matrix. Now, A^T X^ 50 A=A^T X^ 49 (A P A^T ) A =A^T X^ 49 A P (A^T A )=A^T X^ 49 A P =A^T X^ 48 (A P A^T ) A P=A^T X^ 48 A P^2 . =A^T A P^ 50 =I P^ 50 =P^ 50 P= [ array ll 1 & 1 \ 0 & 1 array ] P^2= [ array ll 1 & 2 \ 0 & 1 array ] P^3= [ array ll 1 & 3 \ 0 & 1 array ] .. P^ 50 = [ array cc 1 & 50 \ 0 & 1 array ] So, A^T X^ 50 A=P^ 50 = [ array cc 1 & 50 \ 0 & 1 array ]

Question 5

If A is a square matrix of order 3 , then Adj ( Adj .A^2 ) =

Accepted Answer

| adj ( adj A^2 ) |= | adj A |=| A |^ n-1 =| A |^2 | adj A^2 |=| adj A|^2= (|A|^2 )^2=|A|^4 | adj ( adj A^2 ) |= (|A|^4 )^ 3-1 = (|A|^4 )^2=|A|^8

Question 6

If A= [ array ccc 0 & c & -b \ -c & 0 & a \ b & -a & 0 array ] and B= [ array ccc a^2 & a b & a c \ a b & b^2 & b c \ a c & b c & c^2 array ] , then A B is equal to

Accepted Answer

aligned & A B= [ array ccc 0 & c & -b \ -c & 0 & a \ b & -a & 0 array ] [ array ccc a^2 & a b & a c \ a b & b^2 & b c \ a c & b c & c^2 array ] \ & A B= [ array ccc a b c-a b c & b^2 c-b^2 c & b c^2-b c^2 \ -a^2 c+a^2 c & -a b c+a b c & -a c+a c \ a^2 b-a^2 b & a b^2-a b^2 & a b c-a b c array ] \ & = [ array lll 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 array ]= O \ & aligned

Question 7

If A = [ array cc 0 & 2 \ 3 & -4 array ] and k ~A = [ array cc 0 & 3 a \ 2 b & 24 array ], then the values of k, a and b respectively are:

Accepted Answer

k A= [ array cc 0 & 2 k \ 3 k & -4 k array ]= [ array cc 0 & 3 a \ 2 b & 24 array ] k=-6, a=-4 and b=-9

Question 8

Let A= [ array ccc 1 & -1 & 1 \ 2 & 1 & -3 \ 1 & 1 & 1 array ] and 10 B= [ array ccc 4 & 2 & 2 \ -5 & 0 & \ 1 & -2 & 3 array ] If B is the inverse of A, then the value of is

Accepted Answer

A= [ array ccc 1 & -1 & 1 \ 2 & 1 & -3 \ 1 & 1 & 1 array ] |A|= [ array ccc 1 & -1 & 1 \ 2 & 1 & -3 \ 1 & 1 & 1 array ] =1(1+3)+1(2+3)+1(2-1)=4+5+1=10 Adj A= [ array ccc 4 & -5 & 1 \ 2 & 0 & -2 \ 2 & 5 & 3 array ]^T= [ array ccc 4 & 2 & 2 \ -5 & 0 & 5 \ 1 & -2 & 3 array ] B=A^ -1 = 1 10 [ array ccc 4 & 2 & 2 \ -5 & 0 & 5 \ 1 & -2 & 3 array ] [ A^ -1 = 1 |A| Adj A ] 10 B= [ array ccc 4 & 2 & 2 \ -5 & 0 & 5 \ 1 & -2 & 3 array ] Hence, =5

Question 9

If [ array cc 1 & - \ & 1 array ] [ array cc 1 & \ - & 1 array ]^ -1 = [ array cc a & -b \ b & a array ], then

Accepted Answer

Given, [ array cc 1 & - \ & 1 array ] [ array cc 1 & \ - & 1 array ]^ -1 = [ array cc a & -b \ b & a array ] [ array cc 1 & - \ & 1 array ] 1 1+ ^2 [ array cc 1 & - \ & 1 array ] = [ array cc a & -b \ b & a array ] 1 1+ ^2 [ array cc 1- ^2 & -2 \ 2 & 1- ^2 array ]= [ array cc a & -b \ b & a array ] [ array cc 1- ^2 1+ ^2 & 2 1+ ^2 \ 2 1+ ^2 & 1- ^2 1+ ^2 array ]= [ array cc a & -b \ b & a array ] [ array cc 2 & - 2 \ 2 & 2 array ]= [ array cc a & -b \ b & a array ] a= 2 , b= 2

Matrices — Previous Year Questions & Solutions