I have a matrix $M$ and another $N$. $N$ is an orthogonal (orthogonal => $N^{T} = N^{-1})$ r x r matrix and $M$ is an r x r skew symmetric matrix (skew syemmtric => $M^{T} = -M$). Is $(N^{-1})$$(M^2)$$N$ symmetric? Or is it skew symmetric? Or is it neither of those two options?
My work:
$N$ is orthogonal. So $N^{-1} = N^{T}$. Then we have $(N^{T})$$(M^2)$$N$. $M$ is skew symmetric. So $M^{2} = MM = (-M)(-M) = M^{T}M^{T}$. Then we have $(N^{T})$$(M^{T}M^{T})$$N$. Now what?
$M$ is skew symmetric so $M^T=-M$. Then $$(MM)^T = M^T M^T = (-M)(-M)=MM.$$ That is, $M^2$ is symmetric. From there your $N^{-1}M^2N=N^TM^2N$ is symmetric, too.