The matrix \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*} from Example UM3 is a unitary matrix. By Theorem CUMOS, its columns \begin{equation*} \set{ \colvector{ \frac{1 + i }{{\sqrt{5}}}\\ \frac{1 - i }{{\sqrt{5}}}\\ \frac{i }{{\sqrt{5}}} },\, \colvector{ \frac{3 + 2\,i }{{\sqrt{55}}}\\ \frac{2 + 2\,i }{{\sqrt{55}}}\\ \frac{3 - 5\,i }{{\sqrt{55}}} },\, \colvector{ \frac{2+2i}{\sqrt{22}}\\ \frac{-3 + i }{{\sqrt{22}}}\\ -\frac{2}{\sqrt{22}} } } \end{equation*} form an orthonormal set. You might find checking the six inner products of pairs of these vectors easier than doing the matrix product $\adjoint{U}U$. Or, because the inner product is anti-commutative (Theorem IPAC) you only need check three inner products (see Exercise MINM.T12).