Question

If \[ A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \text{ then } A^{-1} \text{ is:} \]

• A. \( \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \)
• B. \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \)
• C. \( \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
• D. \( \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)

Hint:

For diagonal matrices, the inverse is simply the matrix with reciprocal diagonal elements.

Answer 1

The Correct Option is D

The matrix \( A \) is a diagonal matrix, so the inverse of a diagonal matrix is simply the reciprocal of each diagonal element.
Given: \[ A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] To find \( A^{-1} \), we take the reciprocal of the diagonal elements.
Thus, \[ A^{-1} = \begin{bmatrix} \frac{1}{-1} & 0 & 0 \\ 0 & \frac{1}{1} & 0 \\ 0 & 0 & \frac{1}{1} \end{bmatrix} = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]