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 September 2nd, 2009, 06:31 PM #1 Newbie   Joined: Sep 2009 Posts: 6 Thanks: 0 If A and B are symmetric, show A+B is also symmetric Here's how the problem reads: Let A and B be symmetric matrices. a) Show that A+B is symmetric. b) Show that AB is symmetric if and only if AB=BA.
 September 3rd, 2009, 07:51 AM #2 Senior Member   Joined: Feb 2009 Posts: 172 Thanks: 5 Re: If A and B are symmetric, show A+B is also symmetric a) Since $(A+B)^T=A^T+B^T$, $A=A^T$ and $B=B^T$ you have that $(A+B)^T=A+B$. That is, $A+B$ is symmetric. b) $(\Rightarrow)$ Suppose that $AB$ is symmetric. Then you have that $(AB)^T=AB$. But $(AB)^T=B^TA^T=BA$. Hence, $AB=BA$. $(\Leftarrow)$ Suppose $AB=BA$. Since $(AB)^T=B^TA^T=BA$ and $AB=BA$ you have that $(AB)^T=AB$ that is, $AB$ is symmetric.

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# if a and b are two symmetric matrices, show that ab ba is symmetric

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