If $m,n,r\in \Bbb N$ and $A$ is an $m\times n$ matrix satisfying $(AA^T)^r=I$ is it true that $m=n\implies A$ is invertible.
I think it's true since $\det (AA^T)^r=1\implies \det A^{2r}=1\implies (\det A)^{2r}=1\implies \det A\neq 0\implies A $ is invertible.
So the result is true. Is this correct?