A comment on some recent results concerning the reverse order law for {1, 3, 4}-inverses

This paper has been motivated by the one of Liu and Yang [D. Liu, H. Yang, The reverse order law for {1, 3, 4}-inverse of the product of two matrices, Appl. Math. Comp. 215 (12) (2010) 4293-4303] in which the authors consider separately the cases when (AB){1,3,4}⊆B{1,3,4}·A{1,3,4} and (AB){1,3,4}=B{1,3,4}·A{1,3,4}, where A∈Cn×m and B∈Cm×n. Here we prove that (AB){1,3,4}⊆B{1,3,4}·A{1,3,4} is actually equivalent to (AB){1,3,4}=B{1,3,4}·A{1,3,4}. We show that (AB){1,3,4}⊆B{1,3,4}·A{1,3,4} can only be possible if n⩽m and in this case, we present purely algebraic necessary and sufficient conditions for this inclusion to hold. Also we give some new characterizations of B{1,3,4}·A{1,3,4}⊆(AB){1,3,4}.