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

Answer 1

The correct option is B

$A^2=I$

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