  The main result of this short note is a generic version of Paz's conjecture
on the lengths of generating sets in matrix algebras. Consider a generic
g-tuple A=(A_1,..., A_g) of nxn matrices over a field. We show that whenever
$g^{2d}\geq n^2$, the set of all words of degree 2d in A spans the full nxn
matrix algebra. Our proofs use generic matrices, are combinatorial and depend
on the construction of a special kind of directed multigraphs with few
edge-disjoint walks.
