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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 notgreater than $k$ generate $M_n(\mathbb{F})$ as a vector space. Traditionallythe identity matrix is assumed to be automatically included in all generatingsets $\mathcal S$ and counted as a word of length $0$. In this paper we discusshow the problem changes if this assumption is removed.

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