Find the Inverse of the Following Matrix and Verify that a − 1 a = I 3 ⎡ ⎢ ⎣ 2 3 1 3 4 1 3 7 2 ⎤ ⎥ ⎦ - Mathematics

Question

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

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

Solution

\[B = \begin{bmatrix}2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2\end{bmatrix}\]
Now,
\[ C_{11} = \begin{vmatrix}4 & 1 \\ 7 & 2\end{vmatrix} = 1, C_{12} = - \begin{vmatrix}3 & 1 \\ 3 & 2\end{vmatrix} = - 3\text{ and }C_{13} = \begin{vmatrix}3 & 4 \\ 3 & 7\end{vmatrix} = 9\]
\[ C_{21} = - \begin{vmatrix}3 & 1 \\ 7 & 2\end{vmatrix} = 1, C_{22} = \begin{vmatrix}2 & 1 \\ 3 & 2\end{vmatrix} = 1\text{ and } C_{23} = - \begin{vmatrix}2 & 3 \\ 3 & 7\end{vmatrix} = - 5\]
\[ C_{31} = \begin{vmatrix}3 & 1 \\ 4 & 1\end{vmatrix} = - 1, C_{32} = - \begin{vmatrix}2 & 1 \\ 3 & 1\end{vmatrix} = 1\text{ and }C_{33} = \begin{vmatrix}2 & 3 \\ 3 & 4\end{vmatrix} = - 1\]
\[adjB = \begin{bmatrix}1 & - 3 & 9 \\ 1 & 1 & - 5 \\ - 1 & 1 & - 1\end{bmatrix}^T = \begin{bmatrix}1 & 1 & - 1 \\ - 3 & 1 & 1 \\ 9 & - 5 & - 1\end{bmatrix}\]
\[\text{ and }\left| B \right| = 2\]
\[ B^{- 1} = \frac{1}{2}\begin{bmatrix}1 & 1 & - 1 \\ - 3 & 1 & 1 \\ 9 & - 5 & - 1\end{bmatrix}\]
\[\text{ Now, }B^{- 1} B = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} = I_3\]

shaalaa.com

Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method

Is there an error in this question or solution?

Q 9.2 Q 9.1 Q 10.1

Chapter 7: Adjoint and Inverse of a Matrix - Exercise 7.1 [Page 23]

APPEARS IN

RD Sharma Mathematics [English] Class 12

Chapter 7 Adjoint and Inverse of a Matrix
Exercise 7.1 | Q 9.2 | Page 23

Video TutorialsVIEW ALL [3]

RELATED QUESTIONS

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. School A wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively with a total award money of Rs 1,600. School B wants to spend Rs 2,300 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is Rs 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for an award.

Find the inverse of the matrices (if it exists).

`[(1,0,0),(3,3,0),(5,2,-1)]`

If A = `[(3,1),(-1,2)]` show that A²– 5A + 7I = O. Hence, find A^–1.

If A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` verify that A³ − 6A² + 9A − 4I = O and hence find A⁻¹

Let A = `[(1, sin theta, 1),(-sin theta,1,sin 1),(-1, -sin theta, 1)]` where 0 ≤ θ≤ 2π, then ______.

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.

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.

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.

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.

Find the inverse of the following matrix.

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

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

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.

If \[A = \frac{1}{9}\begin{bmatrix}- 8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & - 8 & 4\end{bmatrix}\],
prove that \[A^{- 1} = A^3\]

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

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

\[\text{ If }A = \begin{bmatrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{bmatrix},\text{ find }A^{- 1}\text{ and show that }A^{- 1} = \frac{1}{2}\left( A^2 - 3I \right) .\]