Answer
The matrices A and B are anti commutative.
Work Step by Step
Calculate $AB$:
$\begin{align}
& AB=\left[ \begin{matrix}
0 & -1 \\
1 & 0 \\
\end{matrix} \right]\left[ \begin{matrix}
1 & 0 \\
0 & -1 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
0\left( 1 \right)-1\left( 0 \right) & 0\left( 0 \right)-1\left( -1 \right) \\
1\left( 1 \right)+0\left( 0 \right) & 1\left( 0 \right)+0\left( -1 \right) \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
0 & 1 \\
1 & 0 \\
\end{matrix} \right]
\end{align}$
Calculate $-BA$:
$\begin{align}
& -BA=-\left[ \begin{matrix}
1 & 0 \\
0 & -1 \\
\end{matrix} \right]\left[ \begin{matrix}
0 & -1 \\
1 & 0 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
-1 & 0 \\
0 & 1 \\
\end{matrix} \right]\left[ \begin{matrix}
0 & -1 \\
1 & 0 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
-1\left( 0 \right)+0\left( 1 \right) & -1\left( -1 \right)+0\left( 0 \right) \\
0\left( 0 \right)+1\left( 1 \right) & 0\left( -1 \right)+1\left( 0 \right) \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
0 & 1 \\
1 & 0 \\
\end{matrix} \right]
\end{align}$
It can be clearly seen that $AB=-BA$.
Hence, the matrices A and B are anti-commutative.
