When Does Rank(A+B)=Rank(A)+Rank(B)?

In a recent note in the Bulletin, Murphy [5] gave a short proof that for complex m×n matrices A and B, r(A+B)=r(A)+r(B) if the rows of A are orthogonal to the rows of B and the columns of A are orthogonal to the columns of B. His proof was elegant and simple, an improvement on an earlier proof of the same result by Meyer [4].