The Forward Order Laws for {1, 2, 3}- and {1, 2, 4}-inverses of a Three Matrix Products

Abstract

In this article, we study the forward order laws for $\{1,2,3\}$- and $\{1,2,4\}$-inverses of a product of
three matrices by using the maximal and minimal ranks of the generalized Schur complement. The necessary and
sufficient conditions for $$A_1\{1,2,3\}A_2\{1,2,3\}A_3\{1,2,3\}\subseteq (A_1A_2A_3)\{1,2,3\}$$ and
$$A_1\{1,2,4\}A_2\{1,2,4\}A_3\{1,2,4\}\subseteq (A_1A_2A_3)\{1,2,4\}$$ are presented.