[Math] Can a product of symmetric matrices give a non-zero skew symmetric matrix

Question

I've been trying to find symmetric $\mathbf{A},\mathbf{B}$ such that $\mathbf{AB}$ is skew-symmetric, but it seems that no matter what I try, I end up forcing $\mathbf{AB}=\mathbf{0}$. Is it possible for this product to be non-zero?

I've also tried proving that $\mathbf{AB}$ must be $\mathbf{0}$ but haven't got much further than $\mathbf{AB}=-\mathbf{BA}$.

Answer 1

Let $$ A=\begin{bmatrix}1&0\\0&-1\end{bmatrix},\ \ \ B=\begin{bmatrix}0&1\\1&0\end{bmatrix}. $$ Then $$ BA=-AB=\begin{bmatrix}0&-1\\1&0\end{bmatrix}. $$