Is it true that for any square matrix of real numbers A, there exists a square matrix B, such that AB is a symmetric matrix?

Is it true that for any square matrix of real numbers $A$, there exists a square matrix $B$, such that $AB$ is a symmetric matrix? This is obviously true if $A$ is invertible, but how about if $A$ is not invertible?

• Yes, it is true. Do you know about transposes of matrices? – hardmath Jan 1 '13 at 15:58
• @hardmath oh, you are right, I forgot about transposes. I think A times A transpose will always be symmetric. – Sunny88 Jan 1 '13 at 16:02
• @Sunny88 Please write that as an answer and accept it. The question will be removed from the unanswered list. – WimC Jan 1 '13 at 16:05

2 Answers

Yes, $AB$ will be symmetric if we let $B=A^{T}$.

• Similarly, $AB$ will be symmetric if $B=SA^T$ for some symmetric matrix $S$. – user1551 Jan 1 '13 at 16:34

You can simply let $B=0$. Then $AB=0$ is symmetric.