# Explain whether this matrix is symmetric or not?

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?

• Then is $((N^{-1})M^{2}N)^{T} = \pm (N^{-1})M^{2}N$, or neither? – Morgan Rodgers Mar 22 at 23:31

$$M$$ 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.