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].