Prove or disprove: $|\det(Q)|=1 \Longrightarrow Q$ is unitary.

Question

I wonder whether the statement of above can be written as an equivalence. So far I could prove the other direction $(\Longleftarrow)$:

If $Q$ is unitary, then

$1=\det(I)=\det(Q^HQ)=\det(Q^H)\det(Q)=\det(Q)\det(Q)=\det(Q)^2 \Longleftrightarrow 1=|\det(Q)|$

However, I can't prove or disprove the other direction mentioned in the question title.

Answer 1

Let

$$Q=\begin{pmatrix} 1/2 & 0 \\ 0 & 2\end{pmatrix}.$$