How to prove that, for $U$ unitary, $|\det U| = 1$ but $\det U\neq \det U^H$?

Question

Prove that unitary matrix $U$ satisfies $|\det U| = 1$, but $\det U$ is different from $\det U^{H}$.

How can I prove these two statements? I guess I should use the fact that every column of unitary matirx is orthonormal, but I'm not sure where to put that...

Answer 1

We have $\det U^H=\overline{\det U}$ and $\det(AB)=\det A\det B$ for two square matrices of the same dimension, which gives $$1=\det I_n=\det U\cdot U^H=\det U\cdot\det U^H=\det U\overline{\det U}=|\det U|^2.$$

We can have $\det U=\det U^H$, when $U=I_n$ for example.