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

Abstract 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 ∈ C n × m and B ∈ C m × 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 } .