<s>[INST] Does $A^TMA = MA^2$ where $A$ is symmetric and $M$ is not necesarily symmetric Let $A, M \in \mathbb{R}^{2\times 2}$, $A$ is symmetric
Does  $A^TMA = MA^2$ hold regardless of whether $M$ is symmetric?
I have tested some simple examples and have not found a violation. Does a counter example exist?
 [/INST] Another counterexample:
$$
\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}
\begin{pmatrix}a & b \\ c & d\end{pmatrix}
\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}
=\begin{pmatrix}4a & 2b \\ 2c & d\end{pmatrix},
$$
but
$$
\begin{pmatrix}a & b \\ c & d\end{pmatrix}
\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}
\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}
=
\begin{pmatrix}a & b \\ c & d\end{pmatrix}
\begin{pmatrix}4 & 0 \\ 0 & 1\end{pmatrix}
=\begin{pmatrix}4a & b \\ 4c & d\end{pmatrix}.
$$
</s>