Classification of matrix subalgebras of length 1

We define the length of a finite system of generators of a given algebra $\mathcal A$ as the smallest number $k$ such that words of length not greater than $k$ generate $\mathcal A$ as a vector space, and the length of the algebra is the maximum of the lengths of its systems of generators. In this paper, we obtain a classification of matrix subalgebras of length 1 up to conjugation. In particular, we describe arbitrary commutative matrix subalgebras of length 1, as well as those that are maximal with respect to inclusion.