#### Answer

Square matrix satisfying the given condition will be $A=\left[ \begin{matrix}
a & 0 \\
0 & a \\
\end{matrix} \right]$. In multiplication of such matrices, simply square each element that is not zero in a square matrix.

#### Work Step by Step

Consider a square matrix of order $2\times 2$:
$A=\left[ \begin{matrix}
a & 0 \\
0 & a \\
\end{matrix} \right]$
Calculate the product AA as below:
$\begin{align}
& AA=\left[ \begin{matrix}
a & 0 \\
0 & a \\
\end{matrix} \right]\left[ \begin{matrix}
a & 0 \\
0 & a \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
a\left( a \right)+0\left( 0 \right) & a\left( 0 \right)+0\left( a \right) \\
0\left( a \right)+a\left( 0 \right) & 0\left( 0 \right)+a\left( a \right) \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
{{a}^{2}} & 0 \\
0 & {{a}^{2}} \\
\end{matrix} \right]
\end{align}$
Therefore, the product of a square matrix to itself, in which each element that is not on the diagonal from upper left to lower right is zero, can be obtained by squaring each element that is not zero in the square matrix.