# Thread: Prove Matrices are symmetric

1. ## Prove Matrices are symmetric

I don't know how to prove this if and only if statement.

Let A & B be symmetric matrices.
Show AB is symmetric if and only if AB=BA.

Any help is much appreciated!

2. You could remember that the product of two symmetric matrices are symmetric iff the matrices commute.

$(AB)^{t}=B^{t}A^{t}=BA$

3. I got that...but I don't know how to express this.

Ok, I know in an if and only if proposition, then you have to prove the statement both ways (forwards and backwards).

So....Assume AB is symmetric. Show AB=BA
(AB)^T=(B^T)(A^T). We know A & B are symmetric so,
AB=BA

Then you have to go the other way...
Assume AB=BA, show AB is symmetric.
So do I do the same this basically as before to show this?
(AB)^T=(B^T)(A^T)=BA

Thanks.