Unclear statement on symmetric/skew symmetric matrices

Question

There's a statement in the textbook that says;

If AB = BA and A and B are symmetric (skew-symmetric), then AB is symmetric.

It looks like it means "symmetric or skew-symmetric". Is there a proof for this?

Answer 1

If $A,B$ are symmetric $(AB)^{T} = (BA)^{T} = A^{T}B^{T} = AB$, where in the first equality we use $AB = BA$ and in the last equality we used the symmetry of $A,B$. If A,B are skew-symmetric this method will give you the same result since $A^{T}B^{T} = (-A)(-B) = AB$