Do two symmetric matrix of the same size commute? [closed]

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If i have two symmetric matrix A and B of the same size, do A and B commute? If not, what is a counter example?

This is related to a problem in my intro to linear alg class.

Thanks.

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2 Answers

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$$\begin{bmatrix}1&1\\1&1\end{bmatrix}\begin{bmatrix}1&0\\0&2\end{bmatrix}=\begin{bmatrix}1&2\\1&2\end{bmatrix}$$ $$\begin{bmatrix}1&0\\0&2\end{bmatrix}\begin{bmatrix}1&1\\1&1\end{bmatrix}=\begin{bmatrix}1&1\\2&2\end{bmatrix}$$

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If $ A = \left( \begin{array}{cc} a & b \\ b & c\\ \end{array} \right) $

and $ B = \left( \begin{array}{cc} x & y \\ y & z\\ \end{array} \right) $

then $AB = \left( \begin{array}{cc} ax+by & ay+bz \\ bx+cy & by+cz\\ \end{array} \right) $

so $BA = \left( \begin{array}{cc} ax + by & bx + cy\\ ay + bz & by + cz\\ \end{array} \right)$.

So, as an example, try taking $A$ with $b=c=0$.

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