Schur complements for tensors and multilinear commutative rank

We show that three notions of rank for matrices of multilinear forms are equivalent. This result generalizes a classical result of Flanders, corrects a minor hole in work of Fortin and Reutenauer, answers a question of Lampert on the relation between the analytic and slice ranks of trilinear forms, and establishes a special case of the conjecture that the analytic and partition ranks of a tensor are equivalent.