  We prove that every coaction of a compact group on a finite-dimensional
$C^*$-algebra is associated with a Fell bundle. Every coaction of a compact
group on a matrix algebra is implemented by a unitary operator. A coaction of a
compact group on $M_n$ is inner if and only if its fixed-point algebra has an
abelian $C^*$-subalgebra of dimension $n$. Investigating the existence of
effective ergodic coactions on $M_n$ reveals that $\operatorname{SO}(3)$ has
them, while $\operatorname{SU}(2)$ does not. We give explicit examples of the
two smallest finite nonabelian groups having effective ergodic coactions on
$M_n$.
