Is $BABA$ a symmetric matrix, if $A = A^T$ and $B = B^T$? Is $BABA$ a symmetric matrix, if $A = A^T$ and $B = B^T$?
I have asked a similar question here, but I am still having doubts (in general).

This is my solution:
We want to show that $BABA = (BABA)^T$.
Starting from the right side, we have that:
$$(BABA)^T = (BA)^T(BA)^T$$
Since we can consider $BA$ as a single term.
Now, we can do the same thing for each $(BA)^T$, so we have:
$$(BA)^T(BA)^T = (A^TB^T)(A^TB^T) = A^TB^TA^TB^T = ABAB$$
Since $A = A^T$ and $B = B^T$.
So, unless $A = B$, we have that the resulting matrix of the multiplication is not symmetric, right?

The answer is NO. Take 
$$A=\left(\begin{array}{cc} 1 & 1 \\ 1 & 1\end{array}\right),
B=\left(\begin{array}{cc} 1 & 0 \\ 0 & 0\end{array}\right)$$.
Then
$$
(BA)^2=BA=\left(\begin{array}{cc} 1 & 1 \\ 0 & 0\end{array}\right)
$$
is not symmetric.
