Question

Let $A=\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]$. The only correct statement about the matrix $A$ is

• A. $A$ is a zero matrix
• B. $A=(-1)I_3$
• C. $A^2=-I$
• D. $A^2=I$

Answer 1

The correct option is D $A^2=I$

$A=\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]$
$A^2=\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\therefore A^2=I$
Hence, option D.