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

## Chapter 7: Adjoint and Inverse of a Matrix

#### Chapter 7: Adjoint and Inverse of a Matrix solutions [Pages 22 - 25]

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.
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.
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.
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.
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.

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.

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.

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.

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.
Q 4 | Page 22

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

Q 5 | Page 23

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

Q 6 | Page 23

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

Q 7.1 | Page 23

Find the inverse of the following matrix:

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

Find the inverse of the following matrix:

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

Find the inverse of the following matrix:

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

Find the inverse of the following matrix:

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

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

Q 8.2 | Page 23

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

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

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

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

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

For the following pair of matrix verity that $\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :$ $A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 6 \\ 3 & 2\end{bmatrix}$

Q 10.2 | Page 23

For the following pair of matrix verity that $\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :$ $A = \begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 5 \\ 3 & 4\end{bmatrix}$

Q 11 | Page 23

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

Q 12 | Page 23

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

Q 13 | Page 23

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

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 $a A^{- 1} = \left( a^2 + bc + 1 \right) I - aA .$

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.

Q 16.1 | 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

$\left[ F \left( \alpha \right) \right]^{- 1} = F \left( - \alpha \right)$
Q 16.2 | 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

$\left[ G \left( \beta \right) \right]^{- 1} = G \left( - \beta \right)$
Q 16.3 | 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

$\left[ G \left( \beta \right) \right]^{- 1} = G \left( - \beta \right)$
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.

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.
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.
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.
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.

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.

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.

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.
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.

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.
Q 27 | Page 24
If $A = \frac{1}{9}\begin{bmatrix}- 8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & - 8 & 4\end{bmatrix}$,
prove that  $A^{- 1} = A^3$
Q 28 | Page 24

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

Q 29 | Page 24

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

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.

Q 31 | Page 24

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

Q 32 | Page 24

Find the matrix X for which

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

Q 33 | Page 24

Find the matrix X satisfying the equation

$\begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix} X \begin{bmatrix}5 & 3 \\ 3 & 2\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} .$
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 $A^2 - 4A - 5I = O$

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

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

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 $A\left( adj A \right) = \left| A \right| I_3$.

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

#### Chapter 7: Adjoint and Inverse of a Matrix solutions [Page 34]

Q 1 | Page 34

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

Q 2 | Page 34

Find the inverse $\begin{bmatrix}5 & 2 \\ 2 & 1\end{bmatrix}$

Q 3 | Page 34

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

Q 4 | Page 34

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

Q 5 | Page 34

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

Q 6 | Page 34

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

Q 7 | Page 34

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

Q 8 | Page 34

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

Q 9 | Page 34

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

Q 10 | Page 34

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

Q 11 | Page 34

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

Q 12 | Page 34

Find the inverse $\begin{bmatrix}1 & 1 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1\end{bmatrix}$

Q 13 | Page 34

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

Q 14 | Page 34

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

Q 15 | Page 34

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

Q 16 | Page 34

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

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

#### Chapter 7: Adjoint and Inverse of a Matrix solutions [Pages 35 - 36]

Q 1 | Page 35

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

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|.

Q 3 | Page 35

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

Q 4 | Page 35

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

Q 5 | Page 35

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

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.

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

Q 8 | Page 35

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

Q 9 | Page 35

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

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).

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).

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.

Q 13 | Page 35

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

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|.

Q 15 | Page 35

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

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.

Q 17 | Page 35

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

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|.

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.

Q 20 | Page 35

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

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.

Q 22 | Page 36

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

Q 23 | Page 36

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

Q 24 | Page 36

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

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).

Q 26 | Page 36

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

Q 27 | Page 36

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

Q 28 | Page 36

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

Q 29 | Page 36

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

Q 30 | Page 36

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

$\begin{bmatrix}2 & 3 \\ 1 & 4\end{bmatrix} \begin{bmatrix}1 & 0 \\ 2 & - 1\end{bmatrix} = \begin{bmatrix}8 & - 3 \\ 9 & - 4\end{bmatrix}$

#### Chapter 7: Adjoint and Inverse of a Matrix solutions [Pages 37 - 39]

Q 1 | Page 37

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) $\left| A \right| \neq 0$

Q 2 | Page 37

If A is an invertible matrix of order 3, then which of the following is not true
(a) $\left| adj A \right| = \left| A \right|^2$
(b) $\left( A^{- 1} \right)^{- 1} = A$
(c) If $BA = CA,\text{ than }B \neq C$ , where B and C are square matrices of order 3
(d) $\left( AB \right)^{- 1} = B^{- 1} A^{- 1} , where B \neq \left[ b_{ij} \right]_{3 \times 3} and \left| B \right| \neq 0$

Q 3 | Page 37

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

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

Q 4 | Page 37

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

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

Q 5 | Page 37

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

Q 6 | Page 37

If A, B are two n × n non-singular matrices, then
(a) AB is non-singular
(b) AB is singular
(c) $\left( AB \right)^{- 1} A^{- 1} B^{- 1}$
(d) (AB)−1 does not exist

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

(a) a27
(b) a9
(c) a6
(d) a2

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

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
(a) Det (A−1)
(b) Det (B−1)
(c) Det (A)
(d) Det (B)

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

(a) 20
(c) 100
(d) 10
(d) 0

Q 11 | Page 37

If A5 = O such that $A^n \neq I\text{ for }1 \leq n \leq 4,\text{ then }\left( I - A \right)^{- 1}$
(a) A4
(b) A3
(c) I + A
(d) none of these

Q 12 | Page 37

If A satisfies the equation $x^3 - 5 x^2 + 4x + \lambda = 0$

(a) $x^3 - 5 x^2 + 4x + \lambda = 0$
(b) $\lambda \neq 2$
(c) $\lambda \neq 2$
(d) $\lambda \neq 0$

Q 13 | Page 37

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

Q 14 | Page 38

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

Q 15 | Page 38

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

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

Q 16 | Page 38

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

Q 17 | Page 38

The matrix $\begin{bmatrix}5 & 10 & 3 \\ - 2 & - 4 & 6 \\ - 1 & - 2 & b\end{bmatrix}$
(a) − 3
(b) 3
(c) 0
(d) non-existent

Q 18 | Page 38

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

Q 19 | Page 38

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

Q 20 | Page 38

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

Q 21 | Page 38

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

Q 22 | Page 38

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

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
(a) $\frac{1}{2}\begin{bmatrix}2 & 4 \\ 3 & - 5\end{bmatrix}$
(b) $\frac{1}{2}\begin{bmatrix}- 2 & 4 \\ 3 & 5\end{bmatrix}$
(c) $\begin{bmatrix}2 & 4 \\ 3 & - 5\end{bmatrix}$
(d) none of these.

Q 24 | Page 38

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

(a) 19
(b) 1/19
(c) − 19
(d) − 1/19

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

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
(b) − A
(c) ab A
(d) none of these

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) $a = 1, b = 1$
(b) $a = \cos 2 \theta, b = \sin 2 \theta$
(c) $a = \sin 2 \theta, b = \cos 2 \theta$
(d) none of these

Q 28 | Page 39

If a matrix A is such that 3

$A^3 + 2 A^2 + 5 A + I = 0,\text{ then }A^{- 1}$
(a) $- \left( 3 A^2 + 2 A + 5 \right)$
(b) $3 A^2 + 2 A + 5$
(c) $3 A^2 - 2 A - 5$
(d) none of these
Q 29 | Page 39

If A is an invertible matrix, then det (A1) is equal to
(a) $\det \left( A \right)$
(b) $\frac{1}{det \left( A \right)}$
(c) 1
(d) none of these

Q 30 | Page 39
If $A = \begin{bmatrix}2 & - 1 \\ 3 & - 2\end{bmatrix},\text{ then } A^n =$

(a) $A = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$, if n is an even natural number
(b) $A = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ , if n is an odd natural number
(c) $A = \begin{bmatrix}- 1 & 0 \\ 0 & 1\end{bmatrix}, if n \in N$
(d) none of these

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
(a) $\begin{bmatrix}x^{- 1} & 0 & 0 \\ 0 & y^{- 1} & 0 \\ 0 & 0 & z^{- 1}\end{bmatrix}$
(b) $xyz \begin{bmatrix}x^{- 1} & 0 & 0 \\ 0 & y^{- 1} & 0 \\ 0 & 0 & z^{- 1}\end{bmatrix}$
(c) $\frac{1}{xyz}\begin{bmatrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{bmatrix}$
(d) $\frac{1}{xyz} \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$

## RD Sharma solutions for Class 12 Mathematics 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 Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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

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