Answer
Yes.
Work Step by Step
$AB=\left[\begin{array}{ll}
0 & -1\\
1 & 0
\end{array}\right]\left[\begin{array}{ll}
1 & 0\\
0 & -1
\end{array}\right]$
$=\left[\begin{array}{ll}
0+0 & 0+1\\
1+0 & 0+0
\end{array}\right]=\left[\begin{array}{ll}
0 & 1\\
1 & 0
\end{array}\right]$
$BA=\left[\begin{array}{ll}
1 & 0\\
0 & -1
\end{array}\right]\left[\begin{array}{ll}
0 & -1\\
1 & 0
\end{array}\right]$
$=\left[\begin{array}{ll}
0+0 & -1+0\\
0-1 & 0+0
\end{array}\right]=\left[\begin{array}{ll}
0 & -1\\
-1 & 0
\end{array}\right]=-AB$
$AB=-BA$
so A and B are anticommutative.