  Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices and let
$\mathcal S$ be a generating set of $M_n(\mathbb{F})$ as an
$\mathbb{F}$-algebra. The length of a finite generating set $\mathcal S$ of
$M_n(\mathbb{F})$ is the smallest number $k$ such that words of length not
greater than $k$ generate $M_n(\mathbb{F})$ as a vector space. Traditionally
the identity matrix is assumed to be automatically included in all generating
sets $\mathcal S$ and counted as a word of length $0$. In this paper we discuss
how the problem changes if this assumption is removed.
