Solution
= Solution
\b[False in general.] For symmetric matrices, $(AB)_{ji}=A_{jk}B_{ki}=B_{ik}A_{kj}=(BA)_{ij}$. The <symmetric product criterion for symmetric matrices> is therefore $AB=BA$. For example,
$$
A=\begin{pmatrix}1&0\\0&0\end{pmatrix},\qquad B=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad AB=\begin{pmatrix}0&1\\0&0\end{pmatrix}
$$
has $(AB)_{12}=1$ and $(AB)_{21}=0$ despite both factors being symmetric.