Length computation of matrix subalgebras of special type

Let $\mathbb F$ be a field and let $\mathcal A$ be a finite-dimensional $\mathbb F$-algebra. We define the length of a finite generating set of this algebra 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 generating sets. In this article, we give a series of examples of length computation for matrix subalgebras. In particular, we evaluate the lengths of certain upper triangular matrix subalgebras and their direct sums, and the lengths of classical commutative matrix subalgebras. The connection between the length of an algebra and the lengths of its subalgebras is also studied.