Question

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

• A. $A=0$
• B. $A=(-1)I$
• C. $A^{-1}$ does not exist
• D. $A^2=I$

Answer 1

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

$A=\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]\therefore A^2=\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]$

$A^2=\left[\begin{array}{ccc}0+0+1& 0& 0\\ 0& 0+1+0& 0\\ 0& 0& 0+0+1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$\therefore A^2=I$

Hence it is a identity matrix