[Math] Product of an orthogonal matrix and a non orthogonal matrix

Question

It's well known that if $U$ and $V$ are orthogonal matrices of dimension $n$, then their product is again an orthogonal matrix. But what can we say about the product of an orthogonal matrix and a non orthogonal matrix? If $P_{n\times n}$ is non orthogonal and $\det(P)\neq \pm 1$, then it is easy to see that the matrices $UP$ and $PU$ can't be orthogonal, since $\det(UP),\det(PU)\neq \pm 1$; what if $P$ is non orthogonal with $\det(P)=\pm 1$?

Answer 1

If $U$ is orthogonal and $UP$ is orthogonal, then $P$ must be orthogonal, since $$ P = U^T(UP) $$ is the product of orthogonal matrices. Similarly, if $U$ and $PU$ are orthogonal, then $P$ must be orthogonal. So: if $U$ is orthogonal and $P$ is non-orthogonal, then $UP$ and $PU$ must be non-orthogonal.

It is possible, however, to have non-orthogonal matrices $P,Q$ such that $PQ$ is orthogonal. Of course, if $P$ is invertible, then $PP^{-1}$ is the identity matrix, which is orthogonal.