An invariance property related to the reverse order law

It is shown that, quite surprisingly, all matrices of the form L−M−, where L− and M− denote generalized inverses of L and M, are generalized inverses of ML if and only if the product MLL−M−ML is invariant with respect to the choice of L− and M−, which at the first glance looks to be a weaker condition than the requirement that MLL−M−ML = ML for every L− and M−. This statement follows as an immediate corollary to the main result of the present note, which provides two criteria for the invariance of expressions of the type KL−M−N involving four given matrices K, L, M, N, with generalized inverses L−, M− of two of them.