Past exam of the mathematical methods I course of the University of Oxford / 2022 / 10 / a / Solution

For unitary matrices, $U^{†}U=I$ and $V^{†}V=I$. The adjoint matrix reverses the order of multiplication, soThus the product is unitary. Likewise $(UV)(UV{)}^{†}=I$, or this follows from invertibility in finite dimension.