Answer
the two matrices are the zero matrix $O$ and the matrix $\left[\begin{array}{cc}
0&1\\
0&0
\end{array}\right]$
Work Step by Step
the two matrices which satisfy that $A^{2}=O$ are the zero matrix $O$ and the matrix $\left[\begin{array}{cc}
0&1\\
0&0
\end{array}\right]$
where $O=\left[\begin{array}{cc}
0&0\\
0&0
\end{array}\right]$
because $O^{2}=O$ and
$\left[\begin{array}{cc}
0&1\\
0&0
\end{array}\right]$ $\left[\begin{array}{cc}
0&1\\
0&0
\end{array}\right]$=$\left[\begin{array}{cc}
0&0\\
0&0
\end{array}\right]=O$
