Question

Let $A=\begin{pmatrix} 0&0 &-1 \\[0.3em] 0 &-1 &0 \\[0.3em] -1 &0&0 \end{pmatrix}$ . The only correct statement about the matrix A is

• A. $A^{-1}$ does not exist
• B. $A = (- 1) I$, where $I$ is a unit matrix
• C. A is a zero matrix
• D. $ A^2 = I.$

Answer 1

Answer: (D)

Clearly A $\neq$ 0
$| A | = -1 \begin{vmatrix} 0&-1 \\[0.3em] -1 &0 \end{vmatrix} = 1 \neq 0 \, \therefore \, A^{-1}$ exists
$A \neq (-1) I \left[-I = \begin{bmatrix}-1&0&0\\ 0&-1&0\\ 0&0&-1\end{bmatrix}\right]$
Again $A^{2} = \begin{bmatrix}0&0&-1\\ 0&-1&0\\ -1&0&0\end{bmatrix}\begin{bmatrix}0&0&-1\\ 0&-1&0\\ -1&0&0\end{bmatrix}$
= $\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix} = I$ .