Question

Do symmetric matrices commute?

Answer 1

symmetric matrices:

No, symmetric matrices do not commute always. If the product of two symmetric matrices is symmetric, then they must commute.

Let, $A=\left[\begin{array}{cc}1& 2\\ 2& 0\end{array}\right],B=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$ are two symmetric matrices.

Then, $AB=\left[\begin{array}{cc}-1& 1\\ 2& -2\end{array}\right]$ is not symmetric.

$BA=\left[\begin{array}{cc}-1& 2\\ 1& -2\end{array}\right]$ is not symmetric

So, $AB\ne BA$

Again, $C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],D=\left[\begin{array}{cc}-2& 1\\ 1& -2\end{array}\right]$are two symmetric matrices.

Then, $CD=\left[\begin{array}{cc}-2& 1\\ 1& -2\end{array}\right]$is symmetric.

$DC=\left[\begin{array}{cc}-2& 1\\ 1& -2\end{array}\right]$is symmetric.

So, $CD=DC$

Therefore, symmetric matrices do not commute always.