\begin{equation*} U= \begin{bmatrix} \frac{1 + i }{{\sqrt{5}}} & \frac{3 + 2\,i }{{\sqrt{55}}} & \frac{2+2i}{\sqrt{22}} \\ \frac{1 - i }{{\sqrt{5}}} & \frac{2 + 2\,i }{{\sqrt{55}}} & \frac{-3 + i }{{\sqrt{22}}} \\ \frac{i }{{\sqrt{5}}} & \frac{3 - 5\,i }{{\sqrt{55}}} & -\frac{2}{\sqrt{22}} \end{bmatrix} \end{equation*} The computations get a bit tiresome, but if you work your way through the computation of $\adjoint{U}U$, you will arrive at the $3\times 3$ identity matrix $I_3$.