Let A and B be two $3\times 3$  matrices of real numbers, where A is symmetric and B is skew symmetric. If  $(A+B)(A-B)=(A-B)(A+B)$  and  ${\left(AB\right)}^T={(-1)}^{k}AB$ , then the possible value(s) of k can be 

 $$\begin{gathered} A\text{'}=A\left(Aissymmetric\right) \\ B\text{'}=-B\left(Bisskewsymmetric\right) \\ (A+B)(A-B)=(A-\mathbf{B})(A+B) \\ \Rightarrow A^2-AB+BA-B^2=A^2+AB-BA-B^2 \end{gathered}$$
$\Rightarrow 2BA=2AB\Rightarrow AB=BA$          ………….(1)
Also,  $\left(AB\right)\text{'}={(-1)}^{k}AB$
 $\Rightarrow A\text{'}B\text{'}={(-1)}^{k}AB$$\Rightarrow -BA={(-1)}^{k}AB$
$\Rightarrow -BA={(-1)}^{k}BA$ using equation (1), we get
k = odd number