The Forward Order Laws for $$\{1,2,3\}$$ { 1 , 2 , 3 } - and $$\{1,2,4\}$$ { 1 , 2 , 4 } -Inverses of Multiple Matrix Products

The forward order law for generalized inverse often appears in linear algebra problems of some applied fields, which have attracted considerable attention and some interesting results have been obtained. In this paper, using the extremal ranks of the generalized Schur complement, we obtain the necessary and sufficient conditions for the forward order laws $$\begin{aligned} A_1\{1,2,3\}A_2\{1,2,3\}\cdots A_n\{1,2,3\}\subseteq (A_1A_2\cdots A_n)\{1,2,3\} \end{aligned}$$ and $$\begin{aligned} A_1\{1,2,4\}A_2\{1,2,4\}\cdots A_n\{1,2,4\}\subseteq (A_1A_2\cdots A_n)\{1,2,4\}. \end{aligned}$$