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

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#### Chapters ## Chapter 7: Adjoint and Inverse of a Matrix

Ex. 7.1Ex. 7.2Ex. 7.3Ex. 7.4

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

Ex. 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.
Ex. 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.
Ex. 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.
Ex. 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.
Ex. 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.

Ex. 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.

Ex. 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.

Ex. 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.

Ex. 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.
Ex. 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.

Ex. 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.

Ex. 7.1 | Q 6 | Page 23

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

Ex. 7.1 | Q 7.1 | Page 23

Find the inverse of the following matrix:

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

Find the inverse of the following matrix:

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

Find the inverse of the following matrix:

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

Find the inverse of the following matrix:

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

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

Ex. 7.1 | Q 8.2 | Page 23

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

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

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

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

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

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

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

Ex. 7.1 | Q 10.2 | Page 23

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

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

Ex. 7.1 | Q 11 | Page 23

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

Ex. 7.1 | Q 12 | Page 23

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

Ex. 7.1 | Q 13 | Page 23

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

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

Ex. 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.

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

Ex. 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.
Ex. 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.
Ex. 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.
Ex. 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.

Ex. 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.

Ex. 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.

Ex. 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.
Ex. 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.

Ex. 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.
Ex. 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  $A^{- 1} = A^3$
Ex. 7.1 | Q 28 | Page 24

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

Ex. 7.1 | Q 29 | Page 24

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

Ex. 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.

Ex. 7.1 | Q 31 | Page 24

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

Ex. 7.1 | Q 32 | Page 24

Find the matrix X for which

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

Ex. 7.1 | Q 33 | Page 24

Find the matrix X satisfying the equation

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

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

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

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

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

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

Ex. 7.2 | Q 1 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 2 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 3 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 4 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 5 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 6 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 7 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 8 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 9 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 10 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 11 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 12 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 13 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 14 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 15 | Page 34

Find the inverse by using elementary row transformations:

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

Ex. 7.2 | Q 16 | Page 34

Find the inverse by using elementary row transformations:

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

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

Ex. 7.3 | Q 1 | Page 35

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

Ex. 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|.

Ex. 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|.

Ex. 7.3 | Q 4 | Page 35

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

Ex. 7.3 | Q 5 | Page 35

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

Ex. 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.

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

Ex. 7.3 | Q 8 | Page 35

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

Ex. 7.3 | Q 9 | Page 35

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

Ex. 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).

Ex. 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).

Ex. 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.

Ex. 7.3 | Q 13 | Page 35

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

Ex. 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|.

Ex. 7.3 | Q 15 | Page 35

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

Ex. 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.

Ex. 7.3 | Q 17 | Page 35

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

Ex. 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|.

Ex. 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.

Ex. 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.

Ex. 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.

Ex. 7.3 | Q 22 | Page 36

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

Ex. 7.3 | Q 23 | Page 36

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

Ex. 7.3 | Q 24 | Page 36

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

Ex. 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).

Ex. 7.3 | Q 26 | Page 36

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

Ex. 7.3 | Q 27 | Page 36

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

Ex. 7.3 | Q 28 | Page 36

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

Ex. 7.3 | Q 29 | Page 36

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

Ex. 7.3 | Q 30 | Page 36

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

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

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

Ex. 7.4 | Q 1 | Page 37

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

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

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

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

• $\left| A \right| \neq 0$

Ex. 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

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

Ex. 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

Ex. 7.4 | Q 4 | Page 37

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

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

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

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

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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 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)

Ex. 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

Ex. 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

Ex. 7.4 | Q 12 | Page 37

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

• $\lambda = 1$

• $\lambda \neq 2$

• $\lambda \neq -1$

• $\lambda \neq 0$

Ex. 7.4 | Q 13 | Page 37

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

• A2

• A3

• A

• none of these

Ex. 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

Ex. 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

Ex. 7.4 | Q 16 | Page 38

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

• $A^{- 1} B C^{- 1}$

• $C^{- 1} B^{- 1} A^{- 1}$

• $CB A^{- 1}$

• $C^{- 1} BA^{- 1}$

Ex. 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

Ex. 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

Ex. 7.4 | Q 19 | Page 38

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

• 1

• 2

• 23

• 26

Ex. 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

Ex. 7.4 | Q 21 | Page 38

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

• $adj A = \left| A \right| A^{- 1}$

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

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

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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 7.1Ex. 7.2Ex. 7.3Ex. 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.

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Concepts covered in Class 12 Maths chapter 7 Adjoint and Inverse of a Matrix are Applications of Determinants and Matrices, Elementary Transformations, Adjoint and Inverse of a Matrix, 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, Area of a Triangle, Minors and Co-factors.

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