The effect of assuming the identity as a generator on the length of the matrix algebra

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.