Answer
See explanation
Work Step by Step
$A^{2}=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right] \cdot\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]=\left[\begin{array}{cc}1+0 & 0-0 \\ 0-0 & 0+1\end{array}\right]=I$
$B^{2}=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] \cdot\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{ll}0+1 & 0+0 \\ 0+0 & 1+0\end{array}\right]=I$
$A B=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right] \cdot\left[\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]$
$B A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] \cdot\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$
So,$A B=-B A$
