Skew-Symmetric vs Symmetric

Question

If $A$ is a symmetric $n × n$ matrix and $B$ is a skew symmetric $n × n$ matrix, which of the following are true?

(a) $ABA$ is symmetric

(b) $ABA$ is skew-symmetric

(c) $AB^2A$ is symmetric

(d) $AB^2A$ is skew-symmetric

I know that b and d holds true.

I am unsure of A and C

However, for a, how does multiplying ABA preserve symmetry, but squaring B preserves symmetry as well?

Answer 1

Use the property $(AB)^t=B^tA^t$ to compute the transpose of each matrix and the fact that $A$ is symmetric and $B$ is skew-symmetric. For example, $(ABA)^t=A^tB^tA^t=A(-B)A=-(ABA)$. Then $ABA$ is skew-symmetric (and not symmetric in general).