Answer
The resulting matrix will have all zeros outside the diagonal,
and on the diagonal, the squares of the corresponding elements
Work Step by Step
Let A be a 2$\times$2 such matrix,
$A=\left[\begin{array}{ll}
a & 0\\
0 & b
\end{array}\right]$
$AA=\left[\begin{array}{ll}
a & 0\\
0 & b
\end{array}\right]\left[\begin{array}{ll}
a & 0\\
0 & b
\end{array}\right]$
$=\left[\begin{array}{ll}
a^{2}+0 & 0+0\\
0+0 & 0+b^{2}
\end{array}\right]$=$\left[\begin{array}{ll}
a^{2} & 0\\
0 & b^{2}
\end{array}\right]$
Let A be a $3\times 3$ such matrix,
$A=\left[\begin{array}{lll}
a & 0 & 0\\
0 & b & 0\\
0 & 0 & c
\end{array}\right]$
$AA=\left[\begin{array}{lll}
a & 0 & 0\\
0 & b & 0\\
0 & 0 & c
\end{array}\right]\left[\begin{array}{lll}
a & 0 & 0\\
0 & b & 0\\
0 & 0 & c
\end{array}\right]$
$=\left[\begin{array}{lll}
a^{2} & 0 & 0\\
0 & b^{2} & 0\\
0 & 0 & c^{2}
\end{array}\right]$
Conjecture:
The resulting matrix will have all zeros outside the diagonal,
and on the diagonal, the squares of the corresponding elements.
