If the multiplication of matrices symmetric matrix, then they can be diagonalized

Question

If the matrix C is the result of the multiplication of matrices A and B, and if is C symmetric matrix, then the matrices A and B can be diagonalized.

Is this statement always correct? If so, why?

Answer 1

Let $$ A=\begin{pmatrix} 1 & 1 \cr 0 & 1 \end{pmatrix},\quad B=A^{-1}. $$ Then $C=I$ is symmetric, but $A$ and $B$ are not diagionalizable.

Answer 2

If $A=B=\left[\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right]$, then $AB$ is the null matrix, which is symmetric. But $A$ is not diagonalizable.