Question

If A and B are symmetric matrices, prove that AB-BA is a skew symmetric matrix

Answer 1

If AB is symmetric, then
AB = (AB)
T = B
TA
T = BA,
which means that A and B commute. On the other hand, if A and B
commute, then
AB = BA = B
TA
T = (AB)
T
,
so that AB is symmetric. Thus AB is symmetric if and only if A and B
commute.
I hope it is clear for you. If you have any doubts please ask me.