Irredundant generating sets for matrix algebras

Let F be a field. We show that the largest irredundant generating sets for the algebra of n×n matrices over F have 2n−1 elements when n>1. (A result of Laffey states that the answer is 2n−2 when n>2, but its proof contains an error.) We further give a classification of the largest irredundant generating sets when n∈{2,3} and F is algebraically closed. We use this description to compute the dimension of the variety of (2n−1)-tuples of n×n matrices which form an irredundant generating set when n∈{2,3}, and draw some consequences to locally redundant generation of Azumaya algebras. In the course of proving the classification, we also determine the largest sets S of subspaces of F³ with the property that every V∈S admits a matrix stabilizing every subspace in S−{V} and not stabilizing V.