Solution
= Solution
A <unitary matrix> has $A^*A=I$. For example,
$$
\boxed{A=\begin{pmatrix}0&i\\i&0\end{pmatrix},\qquad A^*A=I.}
$$
This example is invertible, has nonreal off-diagonal entries, and preserves the standard <Hermitian inner product>. Although it is also symmetric, that additional property is not required for <unitarity>.