On the Rank of the Sum of two Rectangular Matrices

The purpose of this note is to present a short proof for the following theorem. Let A and B be two complex m x n matrices. If B*A = 0 and AB* = 0 then rank(A + B) = rank(A) + rank(B). Let A† and B† be the generalized inverses of A and B, respectively, in the sense of Penrose [ 1]. Now,