The matrix $A=\begin{bmatrix}2 & 0 &0 \\0 & -1 & 0\\0 & 0 & 2\end{bmatrix}$ then $A^{-1}$ is equal to :

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

The correct answer is Option (2) → $\begin{bmatrix}\frac{1}{2} & 0 &0 \\0 & -\frac{1}{1} & 0\\0 & 0 & \frac{1}{2}\end{bmatrix}$ ##

Given matrix:

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

This is a diagonal matrix. For any diagonal matrix, the inverse is obtained by taking the reciprocal of each non-zero diagonal element.

So,

$2 \rightarrow \frac{1}{2}, \quad -1 \rightarrow -1, \quad 2 \rightarrow \frac{1}{2}$

Thus,

$A^{-1} = \begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & \frac{1}{2} \end{bmatrix}$

Verification

$A \cdot A^{-1} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 2 \end{bmatrix} \begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & \frac{1}{2} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I$