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

Answer 1

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

The given matrix $A=\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]$
(A) It is clear that $A$ is not a zero matrix.
(B) $(-1)1=-1\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]\ne A$
$i.e.,(-1)1\ne A$
(C) $\left|A\right|=0\left|\begin{array}{cc}-1& 0\\ 0& 0\end{array}\right|-0\left|\begin{array}{cc}0& 0\\ -1& 0\end{array}\right|-1\left|\begin{array}{cc}0& -1\\ -1& 0\end{array}\right|$
Since $\left|A\right|\ne 0.$ So $A^{-1}$ exists
(D) $A^2=AA\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow A^2=I$