Does multiplying a matrix by it’s transponse always results in a symmetric matrix?
$\displaystyle \begin{array}{cc}a&b\\c&d\end{array}\ $$\displaystyle \
\times\ $$\displaystyle \ \begin{array}{cc}a&c\\b&d\end{array}\ =\ $$\displaystyle \ \begin{array}{cc}a^2+b^2&ac+bd\\ca+db&c^2+d^2\end{ array}\ $$\displaystyle =\ \begin{array}{cc}a^2+b^2&ac+bd\\ac+bd&c^2+d^2\end{ array}$
Which is Symmetric
You can do the same for larger matrices.