Linear Operators which Preserve Pairs on which the Rank is Additive

Let A and B be $m{\times}n$ matrices. A linear operator T preserves the set of matrices on which the rank is additive if rank(A+B) = rank(A)+rank(B) implies that rank(T(A) + T(B)) = rankT(A) + rankT(B). We characterize the set of all linear operators which preserve the set of pairs of $n{\times}n$ matrices on which the rank is additive.