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I am trying to prove this problem and I am having trouble were to exactly start. The question is:

Show that if $A$ is any matrix, then $K = A^T A$ and $L = AA^T$ are both symmetric matrices.

My attempt:

In order to be symmetric then $A=A^T$ then $K = AA$ and since by definition we have that $K = A^n$ is symmetric since $n > 0$.

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    You confuse the variable $A$ in the definition of symmetry with your matrix $A$. Don't do it :D2012-09-21

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You need to use $(XY)^T=Y^TX^T$. Apply this identity and you'll get it.

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    Yup, thank you very much James!2012-09-25