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Question 1:

Find the adjoint of each of the following matrices:

(i) $\left[\begin{array}{cc}-3& 5\\ 2& 4\end{array}\right]$

(ii) $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

(iii)

(iv)

Verify that (adj A) A = |A| I = A (adj A) for the above matrices.

Given below are the squares matrices. Here, we will interchange the diagonal elements and change the signs of
the off-diagonal elements.

s.

Question 2:

Compute the adjoint of each of the following matrices:

(i) $\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$

(ii) $\left[\begin{array}{ccc}1& 2& 5\\ 2& 3& 1\\ -1& 1& 1\end{array}\right]$

(iii) $\left[\begin{array}{ccc}2& -1& 3\\ 4& 2& 5\\ 0& 4& -1\end{array}\right]$

(iv) $\left[\begin{array}{ccc}2& 0& -1\\ 5& 1& 0\\ 1& 1& 3\end{array}\right]$

Verify that (adj AA = |AI = A (adj A) for the above matrices.

Question 3:

For the matrix $A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 3& 0\\ 18& 2& 10\end{array}\right]$, show that A (adj A) = O.

Question 4:

If $A=\left[\begin{array}{ccc}-4& -3& -3\\ 1& 0& 1\\ 4& 4& 3\end{array}\right]$, show that adj A = A.

Question 5:

If $A=\left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right]$, show that adj A = 3AT.

Question 6:

Find A (adj A) for the matrix $A=\left[\begin{array}{ccc}1& -2& 3\\ 0& 2& -1\\ -4& 5& 2\end{array}\right].$

Question 7:

Find the inverse of each of the following matrices:

(i)

(ii) $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

(iii) $\left[\begin{array}{cc}a& b\\ c& \frac{1+bc}{a}\end{array}\right]$

(iv) $\left[\begin{array}{cc}2& 5\\ -3& 1\end{array}\right]$

Question 8:

Find the inverse of each of the following matrices.

(i) $\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\\ 3& 1& 2\end{array}\right]$

(ii) $\left[\begin{array}{ccc}1& 2& 5\\ 1& -1& -1\\ 2& 3& -1\end{array}\right]$

(iii) $\left[\begin{array}{ccc}2& -1& 1\\ -1& 2& -1\\ 1& -1& 2\end{array}\right]$

(iv) $\left[\begin{array}{ccc}2& 0& -1\\ 5& 1& 0\\ 0& 1& 3\end{array}\right]$

(v) $\left[\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]$

(vi) $\left[\begin{array}{ccc}0& 0& -1\\ 3& 4& 5\\ -2& -4& -7\end{array}\right]$

(vii)

Question 9:

Find the inverse of each of the following matrices and verify that .

(i) $\left[\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}\right]$

(ii) $\left[\begin{array}{ccc}2& 3& 1\\ 3& 4& 1\\ 3& 7& 2\end{array}\right]$

Question 10:

For the following pairs of matrices verity that

(i)

(ii)

(AB)1=B1 A1  (AB)1=B1 A1(AB)1=B1 A1

Let

Question 12:

Given $A=\left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]$, compute A−1 and show that $2{A}^{-1}=9I-A.$

Question 13:

If $A=\left[\begin{array}{cc}4& 5\\ 2& 1\end{array}\right]$, then show that

Question 14:

Find the inverse of the matrix $A=\left[\begin{array}{cc}a& b\\ c& \frac{1+bc}{a}\end{array}\right]$ and show that

Question 15:

Given . Compute (AB)−1.

We have,

Let

Show that
(i)
(ii)
(iii) .

​

Question 17:

If $A=\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]$, verify that . Hence, find A−1.

Question 18:

Show that $A=\left[\begin{array}{cc}-8& 5\\ 2& 4\end{array}\right]$ satisfies the equation ${A}^{2}+4A-42I=O$. Hence, find A−1.

Question 19:

If $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$, show that ${A}^{2}-5A+7I=O$.  Hence, find A−1.

Question 20:

If $A=\left[\begin{array}{cc}4& 3\\ 2& 5\end{array}\right]$, find x and y such that ${A}^{2}=xA+yI=O$. Hence, evaluate A−1.

Question 21:

If $A=\left[\begin{array}{cc}3& -2\\ 4& -2\end{array}\right]$, find the value of $\lambda$ so that ${A}^{2}=\lambda A-2I$. Hence, find A−1.

Question 22:

Show that $A=\left[\begin{array}{cc}5& 3\\ -1& -2\end{array}\right]$ satisfies the equation ${x}^{2}-3x-7=0$. Thus, find A−1.

Question 23:

Show that $A=\left[\begin{array}{cc}6& 5\\ 7& 6\end{array}\right]$ satisfies the equation ${x}^{2}-12x+1=O$. Thus, find A−1.

Question 24:

For the matrix $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& -3\\ 2& -1& 3\end{array}\right]$. Show that . Hence, find A−1.

Question 25:

Show that the matrix, $A=\left[\begin{array}{ccc}1& 0& -2\\ -2& -1& 2\\ 3& 4& 1\end{array}\right]$ satisfies the equation, ${A}^{3}-{A}^{2}-3A-{I}_{3}=O$. Hence, find A−1.

Question 26:

If $A=\left[\begin{array}{ccc}2& -1& 1\\ -1& 2& -1\\ 1& -1& 2\end{array}\right]$. Verify that ${A}^{3}-6{A}^{2}+9A-4I=O$ and hence find A−1.

Question 27:

If $A=\frac{1}{9}\left[\begin{array}{ccc}-8& 1& 4\\ 4& 4& 7\\ 1& -8& 4\end{array}\right]$, prove that ${A}^{-1}={A}^{3}$.

Question 28:

If $A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$, show that ${A}^{-1}={A}^{3}$.

Question 29:

If $A=\left[\begin{array}{ccc}-1& 2& 0\\ -1& 1& 1\\ 0& 1& 0\end{array}\right]$, show that ${A}^{2}={A}^{-1}.$

Question 30:

Solve the matrix equation $\left[\begin{array}{cc}5& 4\\ 1& 1\end{array}\right]X=\left[\begin{array}{cc}1& -2\\ 1& 3\end{array}\right]$, where X is a 2 × 2 matrix.

Question 31:

Find the matrix X satisfying the matrix equation

$X\left[\begin{array}{cc}5& 3\\ -1& -2\end{array}\right]=\left[\begin{array}{cc}14& 7\\ 7& 7\end{array}\right]$.

Question 32:

Find the matrix X for which

Question 33:

Find the matrix X satisfying the equation

Question 34:

If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$, find ${A}^{-1}$ and prove that ${A}^{2}-4A-5I=O$

Prove that .

Question 36:

We know that (AB)1 = B1 A1.

Question 37:

If

We know that (AT)1 = (A1)T.

Question 38:

Find the adjoint of the matrix $A=\left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right]$ and hence show that .

Question 1:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{cc}7& 1\\ 4& -3\end{array}\right]$

Question 2:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{cc}5& 2\\ 2& 1\end{array}\right]$

Question 3:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{cc}1& 6\\ -3& 5\end{array}\right]$

Question 4:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]$

Question 5:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{cc}3& 10\\ 2& 7\end{array}\right]$

Question 6:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$

Question 7:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}2& 0& -1\\ 5& 1& 0\\ 0& 1& 3\end{array}\right]$

Question 8:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}2& 3& 1\\ 2& 4& 1\\ 3& 7& 2\end{array}\right]$

Question 9:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$

Question 10:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}1& 2& 0\\ 2& 3& -1\\ 1& -1& 3\end{array}\right]$

Question 11:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}2& -1& 3\\ 1& 2& 4\\ 3& 1& 1\end{array}\right]$

Question 12:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}1& 1& 2\\ 3& 1& 1\\ 2& 3& 1\end{array}\right]$

Question 13:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}2& -1& 4\\ 4& 0& 7\\ 3& -2& 7\end{array}\right]$

Question 14:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}3& 0& -1\\ 2& 3& 0\\ 0& 4& 1\end{array}\right]$

Question 15:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}1& 3& -2\\ -3& 0& -1\\ 2& 1& 0\end{array}\right]$

Question 16:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}-1& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$

Question 17:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right]$

A = IA
$A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right]$

$\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]A\phantom{\rule{0ex}{0ex}}{R}_{2}\to {R}_{2}-2{R}_{1}\phantom{\rule{0ex}{0ex}}\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 1\\ -2& -4& -5\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ -2& 1& 0\\ 0& 0& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}{R}_{3}\to {R}_{3}+2{R}_{1}\phantom{\rule{0ex}{0ex}}\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ -2& 1& 0\\ 2& 0& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}{R}_{2}\to {R}_{2}-{R}_{3}\phantom{\rule{0ex}{0ex}}\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ -4& 1& -1\\ 2& 0& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}$

${R}_{1}\to {R}_{1}-3{R}_{3}\phantom{\rule{0ex}{0ex}}\left[\begin{array}{ccc}1& 2& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}-5& 0& -3\\ -4& 1& -1\\ 2& 0& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}{R}_{1}\to {R}_{1}-2{R}_{2}\phantom{\rule{0ex}{0ex}}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}3& -2& -1\\ -4& 1& -1\\ 2& 0& 1\end{array}\right]$

Therefore,
${A}^{-1}=\left[\begin{array}{ccc}3& -2& -1\\ -4& 1& -1\\ 2& 0& 1\end{array}\right]$

Question 18:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right]$

Let $A=\left[\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right]$

Question 1:

If A is an invertible matrix, then which of the following is not true
(a) ${\left({A}^{2}\right)}^{-1}={\left({A}^{-1}\right)}^{2}$
(b) $\left|{A}^{-1}\right|={\left|A\right|}^{-1}$
(c) ${\left({A}^{T}\right)}^{-1}={\left({A}^{-1}\right)}^{T}$
(d) $\left|A\right|\ne 0$

(a) ${\left({A}^{2}\right)}^{-1}={\left({A}^{-1}\right)}^{2}$

We have,  $\left|{A}^{-1}\right|={\left|A\right|}^{-1}$, ${\left({A}^{T}\right)}^{-1}={\left({A}^{-1}\right)}^{T}$ and $\left|A\right|\ne 0$ all are the properties of the inverse of a matrix A

Question 2:

If A is an invertible matrix of order 3, then which of the following is not true
(a)
(b) ${\left({A}^{-1}\right)}^{-1}=A$
(c) If , where B and C are square matrices of order 3
(d)

(c) If $BA=CA$, then $B\ne C$ where B and C are square matrices of order 3.

If A is an invertible matrix, then ${A}^{-1}$ exists.

Now,
$BA=CA$
On multiplying both sides by ${A}^{-1}$, we get
$BA{A}^{-1}=CA{A}^{-1}$

Therefore, the statement ​given in (c) is not true.

Question 3:

If

(a) is a skew-symmetric matrix
(b) A−1 + B−1
(c) does not exist
(d) none of these

(d) none of these

Question 4:

If $S=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, then adj A is

(a) $\left[\begin{array}{cc}-d& -b\\ -c& a\end{array}\right]$

(b) $\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

(c) $\left[\begin{array}{cc}d& b\\ c& a\end{array}\right]$

(d) $\left[\begin{array}{cc}d& c\\ b& a\end{array}\right]$

(b) $\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Adjoint of a square matrix of order 2 is obtained by interchanging the diagonal elements and changing the signs of off-diagonal elements.

Here,

Question 5:

If A is a singular matrix, then adj A is
(a) non-singular
(b) singular
(c) symmetric
(d) not defined

(b) singular

Question 6:

If A, B are two n × n non-singular matrices, then
(a) AB is non-singular
(b) AB is singular
(c)
(d) (AB)−1 does not exist

(a) AB is non-singular

Question 7:

If $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$, then the value of |adj A| is

(a) a27
(b) a9
(c) a6
(d) a2

(c) a6

Question 8:

If $A=\left[\begin{array}{ccc}1& 2& -1\\ -1& 1& 2\\ 2& -1& 1\end{array}\right]$, then ded (adj (adj A)) is

(a) 144
(b) 143
(c) 142
(d) 14

(a) 144

Question 9:

If B is a non-singular matrix and A is a square matrix, then det (B−1 AB) is equal to
(a) Det (A−1)
(b) Det (B−1)
(c) Det (A)
(d) Det (B)

(c) Det (A)

Question 10:

For any 2 × 2 matrix, if , then |A| is equal to
(a) 20
(c) 100
(d) 10
(d) 0

Question 11:

If A5 = O such that equals
(a) A4
(b) A3
(c) I + A
(d) none of these

Question 12:

If A satisfies the equation ${x}^{3}-5{x}^{2}+4x+\lambda =0$ then A−1 exists if
(a) $\lambda \ne 1$
(b) $\lambda \ne 2$
(c) $\lambda \ne -1$
(d) $\lambda \ne 0$

Question 13:

If for the matrix A, A3 = I, then A−1 =
(a) A2
(b) A3
(c) A
(d) none of these

Question 14:

If A and B are square matrices such that B = − A−1 BA, then (A + B)2 =
(a) O
(b) A2 + B2
(c) A2 + 2AB + B2
(d) A + B

(b) ${A}^{2}+{B}^{2}$

Question 15:

If

(a) 5A
(b) 10A
(c) 16A
(d) 32A

(c) 16A

$A=\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒A=2\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒A=2I\phantom{\rule{0ex}{0ex}}⇒{A}^{5}={\left(2I\right)}^{5}\phantom{\rule{0ex}{0ex}}⇒{A}^{5}=16×2I\phantom{\rule{0ex}{0ex}}⇒{A}^{5}=16\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒{A}^{5}=16A$

Question 16:

For non-singular square matrix A, B and C of the same order
(a)
(b)
(c) $CB{A}^{-1}$
(d)

Disclaimer: In Quesion, We are to find the inverse of . The inverse is missing in the question.

(d)

We have,

Question 17:

The matrix $\left[\begin{array}{ccc}5& 10& 3\\ -2& -4& 6\\ -1& -2& b\end{array}\right]$ is a singular matrix, if the value of b is
(a) − 3
(b) 3
(c) 0
(d) non-existent

(d) non-existent
For any singular matrix, the value of the determinant is 0.
Here,

$A=\left[\begin{array}{ccc}5& 10& 3\\ -2& -4& 6\\ -1& -2& b\end{array}\right]\phantom{\rule{0ex}{0ex}}\left|A\right|=5\left(-4b+12\right)-10\left(-2b+6\right)+3\left(4-4\right)=0\phantom{\rule{0ex}{0ex}}⇒-20b+60+20b-12=0$

Hence, b is non-existent.

Question 18:

If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is
(a) dn
(b) dn−1
(c) dn+1
(d) d

(b) dn−1

We know,
$\left|\mathrm{adj}A\right|={\left|A\right|}^{n-1}$
$⇒\left|\mathrm{adj}A\right|={d}^{n-1}$

Question 19:

If A is a matrix of order 3 and |A| = 8, then |adj A| =
(a) 1
(b) 2
(c) 23
(d) 26

(d) ${2}^{6}$

Question 20:

If ${A}^{2}-A+I=0$, then the inverse of A is
(a) A2
(b) A + I
(c) IA
(d) AI

(c) IA

Question 21:

If A and B are invertible matrices, which of the following statement is not correct.
(a)
(b)
(c) ${\left(A+B\right)}^{-1}={A}^{-1}+{B}^{-1}$
(d)

(c) ${\left(A+B\right)}^{-1}={A}^{-1}+{B}^{-1}$

We have, , and all are the properites of inverse of a matrix.

Question 22:

If A is a square matrix such that A2 = I, then A1 is equal to
(a) A + I
(b) A
(c) 0
(d) 2A

(b) A

$\mathrm{Given}: {A}^{2}=I\phantom{\rule{0ex}{0ex}}$
On multiplying both sides by ${A}^{-1}$ , we get

${A}^{-1}{A}^{2}={A}^{-1}I\phantom{\rule{0ex}{0ex}}⇒A={A}^{-1}I\phantom{\rule{0ex}{0ex}}⇒A={A}^{-1}$

Question 23:

Let and X be a matrix such that A = BX, then X is equal to

(a) $\frac{1}{2}\left[\begin{array}{cc}2& 4\\ 3& -5\end{array}\right]$

(b) $\frac{1}{2}\left[\begin{array}{cc}-2& 4\\ 3& 5\end{array}\right]$

(c) $\left[\begin{array}{cc}2& 4\\ 3& -5\end{array}\right]$

(d) none of these.

(a) $\frac{1}{2}\left[\begin{array}{cc}2& 4\\ 3& -5\end{array}\right]$

Question 24:

If $A=\left[\begin{array}{cc}2& 3\\ 5& -2\end{array}\right]$ be such that ${A}^{-1}=kA$, then k equals
(a) 19
(b) 1/19
(c) − 19
(d) − 1/19

(b) 1/19

Question 25:

If $A=\frac{1}{3}\left[\begin{array}{ccc}1& 1& 2\\ 2& 1& -2\\ x& 2& y\end{array}\right]$ is orthogonal, then x + y =
(a) 3
(b) 0
(c) − 3
(d) 1

Question 26:

If equals
(a) A
(b) − A
(c) ab A
(d) none of these

(d) none of these

Question 27:

If , then

(a)
(b)
(c)
(d) none of these

(b)

Question 28:

If a matrix A is such that 3 equal to
(a)
(b)
(c)
(d) none of these

(d) none of these

$3{A}^{3}+2{A}^{2}+5A+I=0\phantom{\rule{0ex}{0ex}}⇒3{A}^{3}+2{A}^{2}+5A=-I\phantom{\rule{0ex}{0ex}}⇒{A}^{-1}\left(3{A}^{3}+2{A}^{2}+5A\right)=-I{A}^{-1}\phantom{\rule{0ex}{0ex}}⇒3{A}^{2}+2A+5I=-{A}^{-1}\phantom{\rule{0ex}{0ex}}⇒{A}^{-1}=-3{A}^{2}-2A-5I\phantom{\rule{0ex}{0ex}}$

Question 29:

If A is an invertible matrix, then det (A1) is equal to
(a)
(b)
(c) 1
(d) none of these

(b)

We know that for any invertible matrix A, $\left|{A}^{-1}\right|$ = $\frac{1}{\left|A\right|}$.

Question 30:

If

(a) $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, if n is an even natural number

(b) $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, if n is an odd natural number

(c)

(d) none of these

Disclaimer: In all option, the power of A (i.e. n is missing)

(a) ${A}^{n}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, if n is an even natural number

$A=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]\phantom{\rule{0ex}{0ex}}{A}^{2}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒A×A=I\phantom{\rule{0ex}{0ex}}⇒{A}^{-1}=A\phantom{\rule{0ex}{0ex}}$

Generally,

Question 31:

If x, y, z are non-zero real numbers, then the inverse of the matrix $A=\left[\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right]$, is
(a) $\left[\begin{array}{ccc}{x}^{-1}& 0& 0\\ 0& {y}^{-1}& 0\\ 0& 0& {z}^{-1}\end{array}\right]$

(b)

(c) $\frac{1}{xyz}\left[\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right]$

(d)

(a) $\left[\begin{array}{ccc}{x}^{-1}& 0& 0\\ 0& {y}^{-1}& 0\\ 0& 0& {z}^{-1}\end{array}\right]$

Question 32:

If A and B are invertible matrices, then which one of the following is not correct?
(a) adj A$\left|A\right|$ A-1                         (b) det (A-1) = [det(A)]-1
(c) (AB)-1 = B-1A-1                       (d) (A+B)-1 = B-1 + A-1

(a) adj A = $\left|A\right|$ A1

As we know,

Thus, adj A = $\left|A\right|$ A1 is correct.

(b) det(A1) = [det(A)]1

As we know,
$\left|{A}^{-1}\right|=\frac{1}{\left|A\right|}\phantom{\rule{0ex}{0ex}}⇒\left|{A}^{-1}\right|={\left|A\right|}^{-1}\phantom{\rule{0ex}{0ex}}$

Thus, det(A1) = [det(A)]1 is correct.

(c) (AB)1 = B1A1

As we know,
By reversal law of inverse
(AB)1 = B1A1

Thus, (AB)1 = B1A1 is correct.

(d) (A + B)1 = B1 + A1

Thus, (A + B)1 = B1 + A1 is incorrect.

Hence, the correct option is (d).

Question 33:

If A = $\left[\begin{array}{ccc}2& \lambda & -3\\ 0& 2& 5\\ 1& 1& 3\end{array}\right]$, then A-1 exists if
(a) λ=2           (b) λ≠2            (c) λ≠-2            (d) none of these

Given: A = $\left[\begin{array}{ccc}2& \lambda & -3\\ 0& 2& 5\\ 1& 1& 3\end{array}\right]$

A1 exists only if |A| ≠ 0.
$\left|\begin{array}{ccc}2& \lambda & -3\\ 0& 2& 5\\ 1& 1& 3\end{array}\right|\ne 0\phantom{\rule{0ex}{0ex}}⇒2\left(6-5\right)+1\left(5\lambda +6\right)\ne 0\phantom{\rule{0ex}{0ex}}⇒2\left(1\right)+5\lambda +6\ne 0\phantom{\rule{0ex}{0ex}}⇒2+5\lambda +6\ne 0\phantom{\rule{0ex}{0ex}}⇒5\lambda +8\ne 0\phantom{\rule{0ex}{0ex}}⇒5\lambda \ne -8\phantom{\rule{0ex}{0ex}}⇒\lambda \ne \frac{-8}{5}$

Thus, ​A1 exists if λ ∈ $R-\left\{-\frac{8}{5}\right\}$

Hence, the correct option is (a).

Question 1:

If A is a unit matrix of order n, then A (adj A) = ___________________.

As we know that, A(adj A) = |A|I.

But it is given that A is a unit matrix of order n
Therefore, A(adj A) = |I|I = (1)I = I

Hence, if A is a unit matrix of order n, then A (adj A) = I.

Question 2:

If A is a non-singular square matrix such that A3 = I, then A-1 = _______________.

Given: A3 = I

A
3 = I
Multiplying both sides by A1, we get
⇒ A3A1I A1
⇒ A2(AA1) = I A1
⇒ A2(I) = A1
⇒ A2 = A1

Hence, if A is a non-singular square matrix such that A3 = I, then A1 = A2.

Question 3:

If A and B are square matrices of the same order and AB = 3I, then A-1 = __________________.

Given:
A and B are square matrices of the same order
AB = 3I

AB = 3I
Pre-Multiplying both sides by A1, we get
⇒ A1(AB) = A1(3I
(A1A)B = 3(A1I
⇒ (I)B = 3A1
⇒ B = 3A1
⇒ $\frac{1}{3}B$ = A1

Hence, if A and B are square matrices of the same order and AB = 3I, then A=  $\overline{)\frac{1}{3}B}$.

Question 4:

If the matrix A$\left[\begin{array}{ccc}1& a& 2\\ 1& 2& 5\\ 2& 1& 1\end{array}\right]$ is not invertible, than a = ___________________.

Given: A = $\left[\begin{array}{ccc}1& a& 2\\ 1& 2& 5\\ 2& 1& 1\end{array}\right]$

A is not invertible if |A| = 0.
$\left|\begin{array}{ccc}1& \alpha & 2\\ 1& 2& 5\\ 2& 1& 1\end{array}\right|=0\phantom{\rule{0ex}{0ex}}⇒1\left(2-5\right)-1\left(\alpha -2\right)+2\left(5\alpha -4\right)=0\phantom{\rule{0ex}{0ex}}⇒1\left(-3\right)-1\alpha +2+10\alpha -8=0\phantom{\rule{0ex}{0ex}}⇒-3-\alpha +2+10\alpha -8=0\phantom{\rule{0ex}{0ex}}⇒9\alpha -9=0\phantom{\rule{0ex}{0ex}}⇒9\alpha =9\phantom{\rule{0ex}{0ex}}⇒\alpha =1$

Hence, if the matrix A = $\left[\begin{array}{ccc}1& a& 2\\ 1& 2& 5\\ 2& 1& 1\end{array}\right]$ is not invertible, than a = 1.

Question 5:

If A is a singular matrix, then A (adj A) = ____________________.

As we know that, A(adj A) = |A|I.

But it is given that A is a singular matrix
Thus, |A| = 0.
Therefore, A(adj A) = 0I = O, where O is the zero matrix.

Hence, if A is a singular matrix, then A (adj A) = O.

Question 6:

Let A be a square matrix of order 3 such that $\left|A\right|$ = 11 and B be the matrix of confactors of elements of A. Then, ${\left|B\right|}^{2}$ = ________________.

Given:
A be a square matrix of order 3
$\left|A\right|$ = 11
B be the matrix of cofactors of elements of A

Since, B be the matrix of cofactors of elements of A

As we know,

Hence, ${\left|B\right|}^{2}$ = 14641.

Question 7:

If A is a square matrix of order 2 such that A (adj A) =  =______________.

As we know that, A(adj A) = |A|I.

But it is given that A (adj A) = $\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right]$

Hence, |A| = 10.

Question 8:

If A is an invertible matrix of order 3 and  = ___________________.

Given:
A is an invertible matrix of order 3
$\left|A\right|$ = 3

As we know,

Question 9:

If A is an invertible matrix of order 3 and  = __________________.

Given:
A is an invertible matrix of order 3
$\left|A\right|$ = 5

As we know,

Question 10:

If A is an invertible matrix of order 3 and  =__________________.

Given:
A is an invertible matrix of order 3
$\left|A\right|$ = 4

As we know,

Question 11:

If A = diag (1, 2, 3), then  =________________.

Given:
A = diag (1, 2, 3)
⇒ $\left|A\right|$ = 1 × 2 × 3 = 6

As we know,

Question 12:

If A is a square matrix of order 3 such that  = ________________.

Given:
A is a square matrix of order 3
|A| = $\frac{5}{2}$

As we know,
$\left|{A}^{-1}\right|={\left|A\right|}^{-1}\phantom{\rule{0ex}{0ex}}⇒\left|{A}^{-1}\right|=\frac{1}{\left|A\right|}\phantom{\rule{0ex}{0ex}}⇒\left|{A}^{-1}\right|=\frac{1}{\frac{5}{2}}\phantom{\rule{0ex}{0ex}}⇒\left|{A}^{-1}\right|=\frac{2}{5}\phantom{\rule{0ex}{0ex}}$

Hence, $\left|{A}^{-1}\right|=\overline{)\frac{2}{5}}$.

Question 13:

If A is a square matrix such that A (adj A) = 10I, then $\left|A\right|$ = ____________________.

Given:
A is a square matrix

As we know,

Hence, $\left|A\right|=\overline{)10}$.

Question 14:

Let A be a square matrix of order 3 and B = _________________.

Given:
A is a square matrix of order 3
= |A|A−1
|A| = −5

Now,

Hence, $\left|B\right|=\overline{)25}$.

Question 15:

If k is a scalar and I is a unit matrix of order 3, then adj (kI) = ________________.

Given:
I is a unit matrix of order 3

As we know,

Question 16:

If A and A (adj A) = $\left[\begin{array}{cc}k& 0\\ 0& k\end{array}\right]$, then k = _________________.

Given:
A = $\left[\begin{array}{cc}\mathrm{cos}x& \mathrm{sin}x\\ -\mathrm{sin}x& \mathrm{cos}x\end{array}\right]$
A(adj A) = $\left[\begin{array}{cc}k& 0\\ 0& k\end{array}\right]$

Now,
$A=\left[\begin{array}{cc}\mathrm{cos}x& \mathrm{sin}x\\ -\mathrm{sin}x& \mathrm{cos}x\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left|A\right|=\left|\begin{array}{cc}\mathrm{cos}x& \mathrm{sin}x\\ -\mathrm{sin}x& \mathrm{cos}x\end{array}\right|\phantom{\rule{0ex}{0ex}}⇒\left|A\right|={\mathrm{cos}}^{2}x+{\mathrm{sin}}^{2}x\phantom{\rule{0ex}{0ex}}⇒\left|A\right|=1$

As we know,

Hence, $k=\overline{)1}$.

Question 17:

If A is a non-singular matrix of order 3, then adj (adj A) is equal to ________________.

Given:
A is a non-singular matrix of order 3

As we know,

Question 18:

If A = [aij]2×2, where aij = , then A-1 = ____________________.

Given:
A = [aij]2×2, where aij =

Hence, ${A}^{-1}=\overline{)\frac{1}{9}\left[\begin{array}{cc}0& 3\\ 3& 1\end{array}\right]}.$

Question 19:

If A$\left[\begin{array}{cc}0& 3\\ 2& 0\end{array}\right]$ and A-1 = λ (adj A), then λ = _____________________.

Given:
A = $\left[\begin{array}{cc}0& 3\\ 2& 0\end{array}\right]$

Hence, $\lambda =\overline{)-\frac{1}{6}}$.

Question 20:

If A is a 3×3 non-singular matrix such that AAT = ATA and B = A-1AT, then BBT = ______________.

Given:
A is a 3×3 non-singular matrix
AAT = ATA
B
A−1AT

Hence, BBT = I.

Question 21:

If A and B are two square matrices of the same order such that B = -A-1BA, then (A+B)2 = ______________.

Given:
$B=-{A}^{-1}BA\phantom{\rule{0ex}{0ex}}⇒AB=-A{A}^{-1}BA\phantom{\rule{0ex}{0ex}}⇒AB=-IBA\phantom{\rule{0ex}{0ex}}⇒AB=-BA$

Hence, (A + B)2 = A2 + B2.

Question 22:

If A is a non-singular matrix of order 3×3, then (A3)-1 = _____________.

ans

Question 23:

If A be a square matrix such that , then the order of A is __________________.

Given:
A is a square matrix

As we know,

Hence, the order of is 3.

Question 24:

If A = $\left[\begin{array}{ccc}x& 5& 2\\ 2& y& 3\\ 1& 1& z\end{array}\right]$ ,xyz = 80, 3x + 2y + 10z = 20 and A adj A = kI, then k = _________________.

Given:
A = $\left[\begin{array}{ccc}x& 5& 2\\ 2& y& 3\\ 1& 1& z\end{array}\right]$
xyz = 80
3x + 2y + 10z = 20

Now,

As we know,

Hence, k = 79.

Question 1:

Write the adjoint of the matrix $A=\left[\begin{array}{cc}-3& 4\\ 7& -2\end{array}\right].$

Question 2:

If A is a square matrix such that A (adj A)  5I, where I denotes the identity matrix of the same order. Then, find the value of |A|.

We know

Here,

Question 3:

If A is a square matrix of order 3 such that |A| = 5, write the value of |adj A|.

For any square matrix of order n,

Question 4:

If A is a square matrix of order 3 such that |adj A| = 64, find |A|.

For any square matrix of order n,

Question 5:

If A is a non-singular square matrix such that |A| = 10, find |A−1|.

For any non-singular matrix A,

Question 6:

If A, B, C are three non-null square matrices of the same order, write the condition on A such that AB = ACB = C.

Consider $AB=AC$.
On multiplying both sides by ${A}^{-1}$, we get

$A{A}^{-1}B=A{A}^{-1}C$

Therefore, the required condition is A must be invertible or $\left|A\right|\ne 0$.

Question 7:

If A is a non-singular square matrix such that ${A}^{-1}=\left[\begin{array}{cc}5& 3\\ -2& -1\end{array}\right]$, then find ${\left({A}^{T}\right)}^{-1}.$

For any invertible matrix A,
$\phantom{\rule{0ex}{0ex}}\left({A}^{T}{\right)}^{-1}=\left({A}^{-1}{\right)}^{T}$

Question 8:

Given:

For any two non-singular matrices,

Question 9:

If A is symmetric matrix, write whether AT is symmetric or skew-symmetric.

For any symmetric matrix, ${A}^{T}=A$.

Hence, ${A}^{T}$ is also symmetric.

Question 10:

If A is a square matrix of order 3 such that |A| = 2, then write the value of adj (adj A).

For any square matrix A, we have

Question 11:

If A is a square matrix of order 3 such that |A| = 3, then write the value of adj (adj A).

For any square matrix A, we have

Question 12:

If A is a square matrix of order 3 such that adj (2A) = k adj (A), then write the value of k.

Question 13:

If A is a square matrix, then write the matrix adj (AT) − (adj A)T.

Question 14:

Let A be a 3 × 3 square matrix, such that A (adj A) = 2 I, where I is the identity matrix. Write the value of |adj A|.

Question 15:

If A is a non-singular symmetric matrix, write whether A−1 is symmetric or skew-symmetric.

Question 16:

If , then find the value of k.

Question 17:

If A is an invertible matrix such that |A−1| = 2, find the value of |A|.

Question 18:

If A is a square matrix such that , then write the value of |adj A|.

Question 19:

If $A=\left[\begin{array}{cc}2& 3\\ 5& -2\end{array}\right]$ be such that then find the value of k.

Question 20:

Let A be a square matrix such that ${A}^{2}-A+I=O$, then write ${A}^{-1}$ interms of A.

Question 21:

If Cij is the cofactor of the element aij of the matrix $A=\left[\begin{array}{ccc}2& -3& 5\\ 6& 0& 4\\ 1& 5& -7\end{array}\right]$, then write the value of a32C32.

In the given matrix $A=\left[\begin{array}{ccc}2& -3& 5\\ 6& 0& 4\\ 1& 5& -7\end{array}\right]$,
C32 = (−1)3 + 2 (8 − 30) = 22

Therefore, a32C32 = 5 × 22 = 110.

Hence, the value of a32C32 is 110.

Question 22:

Find the inverse of the matrix $\left[\begin{array}{cc}3& -2\\ -7& 5\end{array}\right].$

Question 23:

Find the inverse of the matrix .

Question 24:

If $A=\left[\begin{array}{cc}1& -3\\ 2& 0\end{array}\right]$, write adj A.

Question 26:

If $A=\left[\begin{array}{cc}3& 1\\ 2& -3\end{array}\right]$, then find |adj A|.

Question 27:

If $A=\left[\begin{array}{cc}2& 3\\ 5& -2\end{array}\right]$, write ${A}^{-1}$ in terms of A.

$⇒{A}^{-1}=\frac{1}{19}A$

Write

Question 29:

Use elementary column operation C2 → C2 + 2C1 in the following matrix equation :

Applying C2 → C2 + 2C1, we get

Question 30:

In the following matrix equation use elementary operation R2 → R2 + R1 and the equation thus obtained:

By applying elementary operation R2 → R2 + R1, we get

(Every row operation is equivalent to left-multiplication by an elementary matrix.)

Question 31:

If A is a square matrix with $\left|A\right|$ = 4 then find the value of

Given:
A is a square matrix
$\left|A\right|$ = 4

Now,

Hence,

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