6!

It is given that A and B are symmetric matrices. Therefore, we have:

Thus, ( AB − BA ) is a skew-symmetric matrix.
$$\begin{gathered} \mathrm{and}, \\ \mathrm{We} \mathrm{have}, \\ \left(\mathrm{AB}+\mathrm{BA}\right)\text{'}=\left(\mathrm{AB}\right)\text{'}+\left(\mathrm{BA}\right)\text{'} \\ =\mathrm{B}\text{'}\mathrm{A}\text{'}+\mathrm{A}\text{'}\mathrm{B}\text{'} \\ =\mathrm{BA}+\mathrm{AB} \\ =\mathrm{AB}+\mathrm{BA} \\ \mathrm{So}, \mathrm{AB}+\mathrm{BA} \mathrm{is} \mathrm{a} \mathrm{symmetric} \mathrm{matrix} \end{gathered}$$
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