Do unitary matrices commute?

Rotations in $\mathbb R^3$ do not commute. Consider $$A=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \\ \end{pmatrix}$$

and $$B=\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$$

$A$ is a rotation about the $x$ axis and $B$ is a rotation about the $z$ axis. I will leave it to you to verify that each has determinant $1$ and $AB\ne BA$.