Forward Order Law for the Reflexive Inner Inverse of Multiple Matrix Products ^†

The generalized inverse has numerous important applications in aspects of the theoretic research of matrices and statistics. One of the core problems of generalized inverse is finding the necessary and sufficient conditions for the reverse (or the forward) order laws for the generalized inverse of matrix products. In this paper, by using the extremal ranks of the generalized Schur complement, some necessary and sufficient conditions are given for the forward order law for $A_1\{1,2\}{A}_2\{1,2\}\dots A_n\{1,2\}\subseteq (A_1{A}_2\dots A_n)\{1,2\}$.