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

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.