Lengths of Families of Matrices Allowing a Minimal Polynomial of Given Degree

Let S be a family generating Mat(n)(F) as an algebra over a field F. If dim F[A] = delta > 1 with some A is an element of S, then the products of length at most (2n-delta) & centerdot; [n/delta]-([n/delta](2)-1) & centerdot; (delta + 1) + n-2 & centerdot; (2 delta-1)/n], with multipliers in S, contain the full algebra Mat(n)(F) in their F-linear span.