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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?

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Yes, it is true. Do you know about transposes of matrices? –  hardmath Jan 1 '13 at 15:58
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@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
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@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
    
@WimC I wrote the answer but I can only accept it after two days. –  Sunny88 Jan 1 '13 at 16:10
    
+1 Should also do the job. ;-) –  WimC Jan 1 '13 at 16:14
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2 Answers

up vote 3 down vote accepted

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

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Similarly, $AB$ will be symmetric if $B=SA^T$ for some symmetric matrix $S$. –  user1551 Jan 1 '13 at 16:34
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You can simply let $B=0$. Then $AB=0$ is symmetric.

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