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!

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- Jan 28th 2009, 04:35 PMdude15129Prove 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! - Jan 28th 2009, 04:56 PMgalactus
You could remember that the product of two symmetric matrices are symmetric iff the matrices commute.

$\displaystyle (AB)^{t}=B^{t}A^{t}=BA$ - Jan 28th 2009, 05:12 PMdude15129
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.