If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix, Then what about $A^pB^qA^p$?

Question

A matrix like $A$ is called symmetric iff $A^T=A$.
A matrix like $A$ is called skew-symmetric iff $A^T=-A$.

Suppose that We have two matrices like $A$ which is symmetric and $B$ which is skew-symmetric.
What can we say about $C=A^pB^qA^p$? ( $p,q\ge1)$

Note: I know the answer. $C$ is symmetric if $q$ is even and skew-symmetric if $q$ is odd. But i don't know how to get to the answer. For example i wrote an arbitrary symmetric matrix like $S_{2x2}$ and calculated $S^2,S^3, \dots$ and it seems that they are symmetric too. But that's just an example not a proof.

Thanks in advance.

Answer 1

Hint: Apply the rule $(AB)^T=B^T A^T$ to the product of matrices.