Question

If $A=\begin{bmatrix}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{bmatrix}$, then the only correct statement about the matrix $A$ is

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

Answer 1

Answer: (D)

Now, $A^2=\begin{bmatrix}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{bmatrix}$
$\Rightarrow A^2=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$
$\Rightarrow A^2=1$