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

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Chapter 7: Adjoint and Inverse of a Matrix

Ex. 7.1Ex. 7.10Ex. 7.2Ex. 7.3Ex. 7.4

Chapter 7: Adjoint and Inverse of a Matrix Exercise 7.1, 7.10 solutions [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. 7.10 | 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. 7.10 | 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
\[\text{ If }A = \begin{bmatrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{bmatrix},\text{ find }A^{- 1}\text{ and show that }A^{- 1} = \frac{1}{2}\left( A^2 - 3I \right) .\]

Chapter 7: Adjoint and Inverse of a Matrix Exercise 7.2 solutions [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:

\[\begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]

Chapter 7: Adjoint and Inverse of a Matrix Exercise 7.3 solutions [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:

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

Chapter 7: Adjoint and Inverse of a Matrix Exercise 7.4 solutions [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. 7.10Ex. 7.2Ex. 7.3Ex. 7.4

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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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