Let A be a u by v matrix of rank a, and let M and N be u by g and v by g matrices, respectively, such that M′ AN is nonsingular. Then, rank( A − N( M′ AN) −1 M′ A) = a − g, where g = rank( AN( M′ AN) −1 M′ A) = rank( M′ AN). This is called Wedderburn–Guttman theorem. What happens if M′ AN is rectangular and/or singular? In this paper we investigate conditions under which the regular inverse ( M′ AN) −1 can be replaced by a g-inverse ( M′ AN) − of some kind, thereby extending the Wedderburn–Guttman theorem. The resultant conditions look similar to those arising in seemingly unrelated contexts, namely Cochran’s and related theorems on distributions of quadratic forms involving a normal random vector.