Do unitary matrices commute?

Question

Physically, unitary matrices of the same dimension describes a rigid motion, so it feels like the order of the rigid motion doesn't really matter? So do unitary matrices commute?

Answer 1

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 $AB\ne BA$.

Answer 2

The unitary group $U(n)$ is not abelian for $n\gt1$. The center $Z(U(n))$ is the set of all scalar matrices $\lambda I$, with $\lambda \in U(1)$, by Schur's lemma.