Thread: Symettric Matrices

I was going over some practice problems in my book and I got stuck on this particular one.

It says:

Let A and B be n * n symmetric matrices. Show that AB is symmetric if and only if A and B commute.

Any help would be appreciated!

Let me proof it one way, that should show you how do the other direction.

I shall use "T" to represent the transpose.

By hypothesis A and B are symmetric.
Thus, T(A)=A and T(B)=B by definition.

We show that if AB is symmetric then AB commute.
That is T(AB)=AB
But by a theorem we know that,
T(AB)=T(B)T(A)
Thus,
T(B)T(A)=AB
But T(B)=B and T(A)=A because they are symmetric.
Thus,
BA=AB
And hence they commute.