Is there unitary matrix with eigenvalues $1$ , $i$ , $-i$ , $1-i$ , $1+i$?

Question

Is there unitary matrix with eigenvalues $1$ , $i$ , $-i$ , $1-i$ , $1+i$?

Since modulus of $1+i$ and $1-i$ is not $1$, there exists no such unitary matrix? However this is a question from a past exam, and that would be way too easy. I feel like I'm missing something.

Answer 1

Your argument is correct. Note that suggested eigenvalues all being of modulus $1$ is also a sufficient condition for a unitary matrix to exist.