  We determine minimal generating sets for matrix monoids over semirings and
semifields. In particular, we produce minimal generating sets for the monoids
of upper triangular and unitriangular matrices over a commutative semiring. We
construct minimal and irredundant generating sets for the monoids of $2 \times
2$ and $3 \times 3$ matrices over the tropical integers, showing that
$M_3({\mathbb{Z}}_\mathrm{max})$ is not finitely generated.
