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

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