Let $U, V \in M_{n\times n}$ be orthogonal matrices. Explain why $UV$ is an orthogonal matrix

Question

So I think this has something to do with $UV$ being invertible and then its inverse would be the transpose of $UV$ but I'm not exactly sure how to explain this properly on an exam.

Answer 1

The inverse of $UV$ is its transpose because $(UV)^{-1} = V^{-1}U^{-1} = V^TU^T = (UV)^T$. So it is an orthogonal matrix.