Question: If $A$ and $B$ are $n \times n$ matrices where $A^{-1} = A^{T}$ and the equality $BA^{T} + AB^{T} = 0$ holds, where $0$ is the $n \times n$ matrix with $0$ in every entry here. Then, $A^{T}B + B^{T}A = 0$ as well.
I am having a difficult time showing this equality. Any help would be appreciated.
From $A^{-1}=A^T$, it follows that $AA^T=A^TA=I$. Then, by multiplying $A^T$ to $BA^T+AB^T=0$ from the left and multiplying $A$ to right, you obtain the identity you want.
$BA^T + AB^T = 0$, because $A^{-1}=A^T$, $BA^{-1} + AB^T = 0$ and after we multiply our equation by $A$ and $A^{-1}$ we get $A^{-1}BA^{-1}A + A^{-1}AB^TA = 0$, so we have that $A^TB + B^TA = 0$.