Assuming that M, N are arbitrary matrices on rings, it is discussed that the necessary and sufficient conditions of the weighted Moore-Penrose inverses with respect to M,N for existences. In addition, it is found that the all solutions of A｛1,3M｝, A｛1,4N｝, A｛1,2,3M｝, A｛1,3M,4N｝ and A｛1,2,3M,4N｝.