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. $A^{-1}$ 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 $\ne$ 0
$|A|=-1\left|\begin{array}{cc}0& -1\\ -1& 0\end{array}\right|=1\ne 0\therefore A^{-1}$ exists
$A\ne (-1)I\left[-I=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]\right]$
Again $A^2=\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]$
= $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=I$ .