If A and B  are  two symmetric matrices then AB + BA is

$$\begin{gathered} {\mathrm{A}}^{\mathrm{T}}=\mathrm{A}, {\mathrm{B}}^{\mathrm{T}}=\mathrm{B} \\ \therefore {(\mathrm{AB}+\mathrm{BA})}^{\mathrm{T}}={\left(\mathrm{AB}\right)}^{\mathrm{T}}+{\left(\mathrm{BA}\right)}^{\mathrm{T}} ={\mathrm{B}}^{\mathrm{T}} {\mathrm{A}}^{\mathrm{T}}+{\mathrm{A}}^{\mathrm{T}} {\mathrm{B}}^{\mathrm{T}} \\ =\mathrm{BA}+\mathrm{AB} \end{gathered}$$$\therefore \mathrm{AB}+\mathrm{BA}$ is a symmetric matrix.