Let A, B, C be 3 $\times$ 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements

(S1) A$^{13}$ B$^{26}$ $-$ B$^{26}$ A$^{13}$ is symmetric

(S2) A$^{26}$ C$^{13}$ $-$ C$^{13}$ A$^{26}$ is symmetric

Then,

$A^{T}=A, B^{T}=-B, C^{T}=-C$ <br/><br/> $$ \begin{aligned} P & =A^{13} B^{26}-B^{26} A^{13} \\\\ P^{T} & =\left(A^{13} B^{26}-B^{26} A^{13}\right)^{T}=\left(A^{13} B^{26}\right)^{T}-\left(B^{26} A^{B}\right)^{T} \\\\ & =\left(B^{26}\right)^{T}\left(A^{13}\right)^{T}-\left(A^{13}\right)^{T}\left(B^{26}\right)^{T} \\\\ & =\left(B^{T}\right)^{26}\left(A^{T}\right)^{13}-\left(A^{T}\right)^{13}\left(A^{T}\right)^{26} \\\\ & =B^{26} A^{13}-A^{13} B^{26}=-\left(A^{13} B^{26}-B^{26} A^{13}\right)=-P \end{aligned} $$ <br/><br/> $P$ is skew-symmetric matrix $\Rightarrow S_{1}$ is false <br/><br/> $Q=A^{26} C^{13}-C^{13} A^{26}=Q^{T}=\left(A^{26} C^{13}-C^{13} A^{26}\right)^{T}$ <br/><br/> $Q=\left(A^{26} C^{13}\right)^{T}-\left(C^{13} A^{26}\right)^{T}=\left(C^{13}\right)^{T}\left(A^{26}\right)^{T}-\left(A^{26}\right)^{T}\left(C^{13}\right)^{T}$ <br/><br/> $=\left(C^{T}\right)^{13}\left(A^{T}\right)^{26}-\left(A^{T}\right)^{26}\left(C^{T}\right)^{13}=-C^{13} A^{26}+A^{26} C^{13}$ <br/><br/> $=A^{26} C^{13}+C^{13} A^{26}$ <br/><br/> $\Rightarrow Q^{T}=Q \Rightarrow Q$ is symmetric matrix $\Rightarrow S_{2}$ is true.