# RD Sharma solutions for Class 12 Maths chapter 7 - Adjoint and Inverse of a Matrix [Latest edition]

## Chapter 7: Adjoint and Inverse of a Matrix

Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4
Exercise 7.1 [Pages 22 - 25]

### RD Sharma solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Exercise 7.1 [Pages 22 - 25]

Exercise 7.1 | Q 1.1 | Page 22

Find the adjoint of the following matrix:
$\begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
Exercise 7.1 | Q 1.2 | Page 22

Find the adjoint of the following matrix:
$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
Exercise 7.1 | Q 1.3 | Page 22

Find the adjoint of the following matrix:
$\begin{bmatrix}\cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
Exercise 7.1 | Q 1.4 | Page 22

Find the adjoint of the following matrix:

$\begin{bmatrix}1 & \tan \alpha/2 \\ - \tan \alpha/2 & 1\end{bmatrix}$
Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
Exercise 7.1 | Q 2.1 | Page 22

Compute the adjoint of the following matrix:
$\begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{bmatrix}$

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

Exercise 7.1 | Q 2.2 | Page 22

Compute the adjoint of the following matrix:

$\begin{bmatrix}1 & 2 & 5 \\ 2 & 3 & 1 \\ - 1 & 1 & 1\end{bmatrix}$

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

Exercise 7.1 | Q 2.3 | Page 22

Compute the adjoint of the following matrix:

$\begin{bmatrix}2 & - 1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & - 1\end{bmatrix}$

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

Exercise 7.1 | Q 2.4 | Page 22

Compute the adjoint of the following matrix:

$\begin{bmatrix}2 & 0 & - 1 \\ 5 & 1 & 0 \\ 1 & 1 & 3\end{bmatrix}$

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

Exercise 7.1 | Q 3 | Page 22

For the matrix

$A = \begin{bmatrix}1 & - 1 & 1 \\ 2 & 3 & 0 \\ 18 & 2 & 10\end{bmatrix}$ , show that A (adj A) = O.
Exercise 7.1 | Q 4 | Page 22

If  $A = \begin{bmatrix}- 4 & - 3 & - 3 \\ 1 & 0 & 1 \\ 4 & 4 & 3\end{bmatrix}$, show that adj A = A.

Exercise 7.1 | Q 5 | Page 23

If $A = \begin{bmatrix}- 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1\end{bmatrix}$ , show that adj A = 3AT.

Exercise 7.1 | Q 6 | Page 23

Find A (adj A) for the matrix  $A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & 2 & - 1 \\ - 4 & 5 & 2\end{bmatrix} .$

Exercise 7.1 | Q 7.1 | Page 23

Find the inverse of the following matrix:

$\begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}$
Exercise 7.1 | Q 7.2 | Page 23

Find the inverse of the following matrix:

$\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$
Exercise 7.1 | Q 7.3 | Page 23

Find the inverse of the following matrix:

$\begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}$
Exercise 7.1 | Q 7.4 | Page 23

Find the inverse of the following matrix:

$\begin{bmatrix}2 & 5 \\ - 3 & 1\end{bmatrix}$
Exercise 7.1 | Q 8.1 | Page 23

Find the inverse of the following matrix.
$\begin{bmatrix}1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2\end{bmatrix}$

Exercise 7.1 | Q 8.2 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}1 & 2 & 5 \\ 1 & - 1 & - 1 \\ 2 & 3 & - 1\end{bmatrix}$
Exercise 7.1 | Q 8.3 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}2 & - 1 & 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2\end{bmatrix}$
Exercise 7.1 | Q 8.4 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{bmatrix}$
Exercise 7.1 | Q 8.5 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}0 & 1 & - 1 \\ 4 & - 3 & 4 \\ 3 & - 3 & 4\end{bmatrix}$
Exercise 7.1 | Q 8.6 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}0 & 0 & - 1 \\ 3 & 4 & 5 \\ - 2 & - 4 & - 7\end{bmatrix}$
Exercise 7.1 | Q 8.7 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}1 & 0 & 0 \\ 0 & \cos \alpha & \sin \alpha \\ 0 & \sin \alpha & - \cos \alpha\end{bmatrix}$
Exercise 7.1 | Q 9.1 | Page 23

Find the inverse of the following matrix and verify that $A^{- 1} A = I_3$

$\begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}$
Exercise 7.1 | Q 9.2 | Page 23

Find the inverse of the following matrix and verify that $A^{- 1} A = I_3$

$\begin{bmatrix}2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2\end{bmatrix}$
Exercise 7.1 | Q 10.1 | Page 23

For the following pair of matrix verify that $\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :$

$A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 6 \\ 3 & 2\end{bmatrix}$

Exercise 7.1 | Q 10.2 | Page 23

For the following pair of matrix verify that $\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :$

$A = \begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 5 \\ 3 & 4\end{bmatrix}$

Exercise 7.1 | Q 11 | Page 23

Let $A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B = \begin{bmatrix}6 & 7 \\ 8 & 9\end{bmatrix} .\text{ Find }\left( AB \right)^{- 1}$

Exercise 7.1 | Q 12 | Page 23

Given $A = \begin{bmatrix}2 & - 3 \\ - 4 & 7\end{bmatrix}$, compute A−1 and show that $2 A^{- 1} = 9I - A .$

Exercise 7.1 | Q 13 | Page 23

If $A = \begin{bmatrix}4 & 5 \\ 2 & 1\end{bmatrix}$ , then show that $A - 3I = 2 \left( I + 3 A^{- 1} \right) .$

Exercise 7.1 | Q 14 | Page 23

Find the inverse of the matrix $A = \begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}$ and show that $a A^{- 1} = \left( a^2 + bc + 1 \right) I - aA .$

Exercise 7.1 | Q 15 | Page 23

Given  $A = \begin{bmatrix}5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1\end{bmatrix}, B^{- 1} = \begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}$ . Compute (AB)−1.

Exercise 7.1 | Q 16 | Page 23

Let
$F \left( \alpha \right) = \begin{bmatrix}\cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{bmatrix}\text{ and }G\left( \beta \right) = \begin{bmatrix}\cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ - \sin \beta & 0 & \cos \beta\end{bmatrix}$

Show that

(i) $\left[ F \left( \alpha \right) \right]^{- 1} = F \left( - \alpha \right)$
(ii) $\left[ G \left( \beta \right) \right]^{- 1} = G \left( - \beta \right)$
(iii) $\left[ F \left( \alpha \right)G \left( \beta \right) \right]^{- 1} = G \left( - \beta \right)F \left( - \alpha \right)$
Exercise 7.1 | Q 17 | Page 23

If $A = \begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix}$ , verify that $A^2 - 4 A + I = O,\text{ where }I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\text{ and }O = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$ . Hence, find A−1.

Exercise 7.1 | Q 18 | Page 24

Show that

$A = \begin{bmatrix}- 8 & 5 \\ 2 & 4\end{bmatrix}$ satisfies the equation $A^2 + 4A - 42I = O$. Hence, find A−1.
Exercise 7.1 | Q 19 | Page 24

If $A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix}$, show that

$A^2 - 5A + 7I = O$.  Hence, find A−1.
Exercise 7.1 | Q 20 | Page 24

If  $A = \begin{bmatrix}4 & 3 \\ 2 & 5\end{bmatrix}$, find x and y such that

$A^2 = xA + yI = O$ . Hence, evaluate A−1.
Exercise 7.1 | Q 21 | Page 24

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

Exercise 7.1 | Q 22 | Page 24

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

Exercise 7.1 | Q 23 | Page 24

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

Exercise 7.1 | Q 24 | Page 24

For the matrix $A = \begin{bmatrix}1 & 1 & 1 \\ 1 & 2 & - 3 \\ 2 & - 1 & 3\end{bmatrix}$ . Show that

$A^{- 3} - 6 A^2 + 5A + 11 I_3 = O$. Hence, find A−1.
Exercise 7.1 | Q 25 | Page 24

Show that the matrix, $A = \begin{bmatrix}1 & 0 & - 2 \\ - 2 & - 1 & 2 \\ 3 & 4 & 1\end{bmatrix}$  satisfies the equation,  $A^3 - A^2 - 3A - I_3 = O$ . Hence, find A−1.

Exercise 7.1 | Q 26 | Page 24
If $A = \begin{bmatrix}2 & - 1 & 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2\end{bmatrix}$.
Verify that $A^3 - 6 A^2 + 9A - 4I = O$  and hence find A−1.
Exercise 7.1 | Q 27 | Page 24
If $A = \frac{1}{9}\begin{bmatrix}- 8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & - 8 & 4\end{bmatrix}$,
prove that  $A^{- 1} = A^3$
Exercise 7.1 | Q 28 | Page 24

If $A = \begin{bmatrix}3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1\end{bmatrix}$ , show that $A^{- 1} = A^3$

Exercise 7.1 | Q 29 | Page 24

If $A = \begin{bmatrix}- 1 & 2 & 0 \\ - 1 & 1 & 1 \\ 0 & 1 & 0\end{bmatrix}$ , show that  $A^2 = A^{- 1} .$

Exercise 7.1 | Q 30 | Page 24

Solve the matrix equation $\begin{bmatrix}5 & 4 \\ 1 & 1\end{bmatrix}X = \begin{bmatrix}1 & - 2 \\ 1 & 3\end{bmatrix}$, where X is a 2 × 2 matrix.

Exercise 7.1 | Q 31 | Page 24

Find the matrix X satisfying the matrix equation $X\begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix} = \begin{bmatrix}14 & 7 \\ 7 & 7\end{bmatrix}$

Exercise 7.1 | Q 32 | Page 24

Find the matrix X for which

$\begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix} X \begin{bmatrix}- 1 & 1 \\ - 2 & 1\end{bmatrix} = \begin{bmatrix}2 & - 1 \\ 0 & 4\end{bmatrix}$

Exercise 7.1 | Q 33 | Page 24

Find the matrix X satisfying the equation

$\begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix} X \begin{bmatrix}5 & 3 \\ 3 & 2\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} .$
Exercise 7.1 | Q 34 | Page 24

If $A = \begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{bmatrix}$ , find $A^{- 1}$ and prove that $A^2 - 4A - 5I = O$

Exercise 7.1 | Q 36 | Page 25
$\text{ If }A^{- 1} = \begin{bmatrix}3 & - 1 & 1 \\ - 15 & 6 & - 5 \\ 5 & - 2 & 2\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 2 & - 2 \\ - 1 & 3 & 0 \\ 0 & - 2 & 1\end{bmatrix},\text{ find }\left( AB \right)^{- 1} .$
Exercise 7.1 | Q 37 | Page 25

If $A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1\end{bmatrix},\text{ find }\left( A^T \right)^{- 1} .$

Exercise 7.1 | Q 38 | Page 25

Find the adjoint of the matrix $A = \begin{bmatrix}- 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1\end{bmatrix}$  and hence show that $A\left( adj A \right) = \left| A \right| I_3$.

Exercise 7.1 | Q 39 | Page 25
$\text{ If }A = \begin{bmatrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{bmatrix},\text{ find }A^{- 1}\text{ and show that }A^{- 1} = \frac{1}{2}\left( A^2 - 3I \right) .$
Exercise 7.2 [Page 34]

### RD Sharma solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Exercise 7.2 [Page 34]

Exercise 7.2 | Q 1 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}7 & 1 \\ 4 & - 3\end{bmatrix}$

Exercise 7.2 | Q 2 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}5 & 2 \\ 2 & 1\end{bmatrix}$

Exercise 7.2 | Q 3 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}1 & 6 \\ - 3 & 5\end{bmatrix}$

Exercise 7.2 | Q 4 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}$

Exercise 7.2 | Q 5 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}3 & 10 \\ 2 & 7\end{bmatrix}$

Exercise 7.2 | Q 6 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}$

Exercise 7.2 | Q 7 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{bmatrix}$

Exercise 7.2 | Q 8 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}2 & 3 & 1 \\ 2 & 4 & 1 \\ 3 & 7 & 2\end{bmatrix}$

Exercise 7.2 | Q 9 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1\end{bmatrix}$

Exercise 7.2 | Q 10 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}1 & 2 & 0 \\ 2 & 3 & - 1 \\ 1 & - 1 & 3\end{bmatrix}$

Exercise 7.2 | Q 11 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}2 & - 1 & 3 \\ 1 & 2 & 4 \\ 3 & 1 & 1\end{bmatrix}$

Exercise 7.2 | Q 12 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}1 & 1 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1\end{bmatrix}$

Exercise 7.2 | Q 13 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}2 & - 1 & 4 \\ 4 & 0 & 7 \\ 3 & - 2 & 7\end{bmatrix}$

Exercise 7.2 | Q 14 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}3 & 0 & - 1 \\ 2 & 3 & 0 \\ 0 & 4 & 1\end{bmatrix}$

Exercise 7.2 | Q 15 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}1 & 3 & - 2 \\ - 3 & 0 & - 1 \\ 2 & 1 & 0\end{bmatrix}$

Exercise 7.2 | Q 16 | Page 34

Find the inverse by using elementary row transformations:

$\begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}$

Exercise 7.3 [Pages 35 - 36]

### RD Sharma solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Exercise 7.3 [Pages 35 - 36]

Exercise 7.3 | Q 1 | Page 35

Write the adjoint of the matrix $A = \begin{bmatrix}- 3 & 4 \\ 7 & - 2\end{bmatrix} .$

Exercise 7.3 | Q 2 | Page 35

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|.

Exercise 7.3 | Q 3 | Page 35

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

Exercise 7.3 | Q 4 | Page 35

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

Exercise 7.3 | Q 5 | Page 35

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

Exercise 7.3 | Q 6 | Page 35

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

Exercise 7.3 | Q 7 | Page 35

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

Exercise 7.3 | Q 8 | Page 35

If adj $A = \begin{bmatrix}2 & 3 \\ 4 & - 1\end{bmatrix}\text{ and adj }B = \begin{bmatrix}1 & - 2 \\ - 3 & 1\end{bmatrix}$

Exercise 7.3 | Q 9 | Page 35

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

Exercise 7.3 | Q 10 | Page 35

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

Exercise 7.3 | Q 11 | Page 35

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

Exercise 7.3 | Q 12 | Page 35

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

Exercise 7.3 | Q 13 | Page 35

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

Exercise 7.3 | Q 14 | Page 35

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|.

Exercise 7.3 | Q 15 | Page 35

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

Exercise 7.3 | Q 16 | Page 35

If $A = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\text{ and }A \left( adj A = \right)\begin{bmatrix}k & 0 \\ 0 & k\end{bmatrix}$, then find the value of k.

Exercise 7.3 | Q 17 | Page 35

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

Exercise 7.3 | Q 18 | Page 35

If A is a square matrix such that $A \left( adj A \right) = \begin{bmatrix}5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{bmatrix}$ , then write the value of |adj A|.

Exercise 7.3 | Q 19 | Page 35

If $A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}$ be such that $A^{- 1} = k A,$  then find the value of k.

Exercise 7.3 | Q 20 | Page 35

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

Exercise 7.3 | Q 21 | Page 36

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

Exercise 7.3 | Q 22 | Page 36

Find the inverse of the matrix $\begin{bmatrix}3 & - 2 \\ - 7 & 5\end{bmatrix} .$

Exercise 7.3 | Q 23 | Page 36

Find the inverse of the matrix $\begin{bmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}$

Exercise 7.3 | Q 24 | Page 36

If $A = \begin{bmatrix}1 & - 3 \\ 2 & 0\end{bmatrix}$, write adj A.

Exercise 7.3 | Q 25 | Page 36

If $A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}, B = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ , find adj (AB).

Exercise 7.3 | Q 26 | Page 36

If $A = \begin{bmatrix}3 & 1 \\ 2 & - 3\end{bmatrix}$, then find |adj A|.

Exercise 7.3 | Q 27 | Page 36

If $A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}$ , write  $A^{- 1}$ in terms of A.

Exercise 7.3 | Q 28 | Page 36

Write $A^{- 1}\text{ for }A = \begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}$

Exercise 7.3 | Q 29 | Page 36

Use elementary column operation C2 → C2 + 2C1 in the following matrix equation : $\begin{bmatrix} 2 & 1 \\ 2 & 0\end{bmatrix} = \begin{bmatrix}3 & 1 \\ 2 & 0\end{bmatrix}\begin{bmatrix}1 & 0 \\ - 1 & 1\end{bmatrix}$

Exercise 7.3 | Q 30 | Page 36

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

$\begin{bmatrix}2 & 3 \\ 1 & 4\end{bmatrix} \begin{bmatrix}1 & 0 \\ 2 & - 1\end{bmatrix} = \begin{bmatrix}8 & - 3 \\ 9 & - 4\end{bmatrix}$
Exercise 7.4 [Pages 37 - 39]

### RD Sharma solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Exercise 7.4 [Pages 37 - 39]

Exercise 7.4 | Q 1 | Page 37

If A is an invertible matrix, then which of the following is not true ?

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

• $\left| A^{- 1} \right| = \left| A \right|^{- 1}$

• $\left( A^T \right)^{- 1} = \left( A^{- 1} \right)^T$

• $\left| A \right| \neq 0$

Exercise 7.4 | Q 2 | Page 37

If A is an invertible matrix of order 3, then which of the following is not true ?

• $\left| adj A \right| = \left| A \right|^2$

• $\left( A^{- 1} \right)^{- 1} = A$

• If $BA = CA,\text{ than }B \neq C$ , where B and C are square matrices of order 3

• $\left( AB \right)^{- 1} = B^{- 1} A^{- 1} , where B \neq \left[ b_{ij} \right]_{3 \times 3} and \left| B \right| \neq 0$

Exercise 7.4 | Q 3 | Page 37

If $A = \begin{bmatrix}3 & 4 \\ 2 & 4\end{bmatrix}, B = \begin{bmatrix}- 2 & - 2 \\ 0 & - 1\end{bmatrix},\text{ then }\left( A + B \right)^{- 1} =$

• is a skew-symmetric matrix

• A−1 + B−1

• does not exist

• none of these

Exercise 7.4 | Q 4 | Page 37

If $S = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$, then adj A is ____________ .

• $\begin{bmatrix}- d & - b \\ - c & a\end{bmatrix}$

• $\begin{bmatrix}d & - b \\ - c & a\end{bmatrix}$

• $\begin{bmatrix}d & b \\ c & a\end{bmatrix}$

• $\begin{bmatrix}d & c \\ b & a\end{bmatrix}$

Exercise 7.4 | Q 5 | Page 37

If A is a singular matrix, then adj A is _____________ .
(a)
(b)
(c)
(d) not defined

• non-singular

• singular

• symmetric

• not defined

Exercise 7.4 | Q 6 | Page 37

If A, B are two n × n non-singular matrices, then __________ .

• AB is non-singular

• AB is singular

• $\left( AB \right)^{- 1} A^{- 1} B^{- 1}$

• (AB)−1 does not exist

Exercise 7.4 | Q 7 | Page 37

If $A = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}$ , then the value of |adj A| is _____________ .

• a27

• a9

• a6

• a2

Exercise 7.4 | Q 8 | Page 37

If $A = \begin{bmatrix}1 & 2 & - 1 \\ - 1 & 1 & 2 \\ 2 & - 1 & 1\end{bmatrix}$ , then ded (adj (adj A)) is __________ .

• 144

• 143

• 142

• 14

Exercise 7.4 | Q 9 | Page 37

If B is a non-singular matrix and A is a square matrix, then det (B−1 AB) is equal to ___________ .

• Det (A−1)

• Det (B−1)

• Det (A)

• Det (B)

Exercise 7.4 | Q 10 | Page 37

For any 2 × 2 matrix, if $A \left( adj A \right) = \begin{bmatrix}10 & 0 \\ 0 & 10\end{bmatrix}$ , then |A| is equal to ______ .

• 20

• 100

• 10

• 0

Exercise 7.4 | Q 11 | Page 37

If A5 = O such that $A^n \neq I\text{ for }1 \leq n \leq 4,\text{ then }\left( I - A \right)^{- 1}$ equals ________ .

• A4

• A3

• I + A

• none of these

Exercise 7.4 | Q 12 | Page 37

If A satisfies the equation $x^3 - 5 x^2 + 4x + \lambda = 0$ then A-1 exists if _____________ .

• $\lambda = 1$

• $\lambda \neq 2$

• $\lambda \neq -1$

• $\lambda \neq 0$

Exercise 7.4 | Q 13 | Page 37

If for the matrix A, A3 = I, then A−1 = _____________ .

• A2

• A3

• A

• none of these

Exercise 7.4 | Q 14 | Page 38

If A and B are square matrices such that B = − A−1 BA, then (A + B)2 = ________ .

• O

• A2 + B2

• A2 + 2AB + B2

• A + B

Exercise 7.4 | Q 15 | Page 38

If $A = \begin{bmatrix}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{bmatrix},\text{ then }A^5 =$ ____________ .

• 5A

• 10A

• 16A

• 32A

Exercise 7.4 | Q 16 | Page 38

For non-singular square matrix A, B and C of the same order $\left( A B^{- 1} C \right) =$ ______________ .

• $A^{- 1} B C^{- 1}$

• $C^{- 1} B^{- 1} A^{- 1}$

• $CB A^{- 1}$

• $C^{- 1} BA^{- 1}$

Exercise 7.4 | Q 17 | Page 38

The matrix $\begin{bmatrix}5 & 10 & 3 \\ - 2 & - 4 & 6 \\ - 1 & - 2 & b\end{bmatrix}$ is a singular matrix, if the value of b is _____________ .

• -3

• 3

• 0

• non-existent

Exercise 7.4 | Q 18 | Page 38

If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is _____________ .

• dn

• dn−1

• dn+1

• d

Exercise 7.4 | Q 19 | Page 38

If A is a matrix of order 3 and |A| = 8, then |adj A| = __________ .

• 1

• 2

• 23

• 26

Exercise 7.4 | Q 20 | Page 38

If $A^2 - A + I = 0$, then the inverse of A is __________ .

• A2

• A + I

• I − A

• A − I

Exercise 7.4 | Q 21 | Page 38

If A and B are invertible matrices, which of the following statement is not correct.

• $adj A = \left| A \right| A^{- 1}$

• $\det \left( A^{- 1} \right) = \left( \det A \right)^{- 1}$

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

• $\left( AB \right)^{- 1} = B^{- 1} A^{- 1}$

Exercise 7.4 | Q 22 | Page 38

If A is a square matrix such that A2 = I, then A1 is equal to _______ .

• A + I

• A

• 0

• 2A

Exercise 7.4 | Q 23 | Page 38

Let $A = \begin{bmatrix}1 & 2 \\ 3 & - 5\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$ and X be a matrix such that A = BX, then X is equal to _____________ .

• $\frac{1}{2}\begin{bmatrix}2 & 4 \\ 3 & - 5\end{bmatrix}$

• $\frac{1}{2}\begin{bmatrix}- 2 & 4 \\ 3 & 5\end{bmatrix}$

• $\begin{bmatrix}2 & 4 \\ 3 & - 5\end{bmatrix}$

• none of these

Exercise 7.4 | Q 24 | Page 38

If $A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}$  be such that $A^{- 1} = kA$, then k equals ___________ .

• 19

• 1/19

• -19

• -1/19

Exercise 7.4 | Q 25 | Page 38
If $A = \frac{1}{3}\begin{bmatrix}1 & 1 & 2 \\ 2 & 1 & - 2 \\ x & 2 & y\end{bmatrix}$ is orthogonal, then x + y =

(a) 3
(b) 0
(c) − 3
(d) 1

• 3

• 0

• -3

• 1

• None of these

Exercise 7.4 | Q 26 | Page 38

If $A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 0 & 1 \\ a & b & 2\end{bmatrix},\text{ then aI + bA + 2 }A^2$ equals ____________ .

• A

• -A

• ab A

• none of these

Exercise 7.4 | Q 27 | Page 38

If $\begin{bmatrix}1 & - \tan \theta \\ \tan \theta & 1\end{bmatrix} \begin{bmatrix}1 & \tan \theta \\ - \tan \theta & 1\end{bmatrix} - 1 = \begin{bmatrix}a & - b \\ b & a\end{bmatrix}$, then _______________ .

• $a = 1, b = 1$

• $a = \cos 2 \theta, b = \sin 2 \theta$

• $a = \sin 2 \theta, b = \cos 2 \theta$

• None of these

Exercise 7.4 | Q 28 | Page 39

If a matrix A is such that $3A^3 + 2 A^2 + 5 A + I = 0,\text{ then }A^{- 1}$ equal to _______________ .

• $- \left( 3 A^2 + 2 A + 5 \right)$

• $3 A^2 + 2 A + 5$

• $3 A^2 - 2 A - 5$

• none of these

Exercise 7.4 | Q 29 | Page 39

If A is an invertible matrix, then det (A1) is equal to ____________ .

• det (A)

• $\frac{1}{det \left( A \right)}$

• 1

• none of these

Exercise 7.4 | Q 30 | Page 39
If $A = \begin{bmatrix}2 & - 1 \\ 3 & - 2\end{bmatrix},\text{ then } A^n =$ ______________ .
• $A^n = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$, if n is an even natural number

• $A^n = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ , if n is an odd natural number

• $A^n = \begin{bmatrix}- 1 & 0 \\ 0 & 1\end{bmatrix}$, if n ∈ N

• none of these

Exercise 7.4 | Q 31 | Page 39
If x, y, z are non-zero real numbers, then the inverse of the matrix $A = \begin{bmatrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{bmatrix}$, is _____________ .
• $\begin{bmatrix}x^{- 1} & 0 & 0 \\ 0 & y^{- 1} & 0 \\ 0 & 0 & z^{- 1}\end{bmatrix}$

• $xyz \begin{bmatrix}x^{- 1} & 0 & 0 \\ 0 & y^{- 1} & 0 \\ 0 & 0 & z^{- 1}\end{bmatrix}$

• $\frac{1}{xyz}\begin{bmatrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{bmatrix}$

• $\frac{1}{xyz} \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$

## Chapter 7: Adjoint and Inverse of a Matrix

Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4

## RD Sharma solutions for Class 12 Maths chapter 7 - Adjoint and Inverse of a Matrix

RD Sharma solutions for Class 12 Maths chapter 7 (Adjoint and Inverse of a Matrix) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12 Maths chapter 7 Adjoint and Inverse of a Matrix are Applications of Determinants and Matrices, Elementary Transformations, Inverse of a Square Matrix by the Adjoint Method, Properties of Determinants, Determinant of a Square Matrix, Determinants of Matrix of Order One and Two, Determinant of a Matrix of Order 3 × 3, Rule A=KB, Introduction of Determinant, Minors and Co-factors, Area of a Triangle.

Using RD Sharma Class 12 solutions Adjoint and Inverse of a Matrix exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.

Get the free view of chapter 7 Adjoint and Inverse of a Matrix Class 12 extra questions for Class 12 Maths and can use Shaalaa.com to keep it handy for your exam preparation