  Necessary and sufficient conditions are given on matrices $A$, $B$ and $S$,
having entries in some field $\mathbb F$ and suitable dimensions, such that the
linear span of the terms $A^iSB^j$ over $\mathbb F$ is equal to the whole
matrix space. This result is then used to determine the cardinality of subsets
of $\mathbb F[A]S\mathbb F[B]$ when $\mathbb F$ is a finite field.

Rings and Algebras