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!

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- Feb 24th 2007, 03:02 PMbuckaroobillSymettric 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! - Feb 24th 2007, 03:16 PMThePerfectHacker
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.