Thread: Symettric Matrices

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.