Rank equalities related to the generalized inverse AT,S(2) with applications

A variety of rank formulas of some matrix expressions and certain partitioned matrices with respect to the generalized inverse AT,S(2) are established. Some necessary and sufficient conditions are given by using the rank formulas presented in this paper for two, three and four ordered matrices to be independent in the generalized inverse AT,S(2). As special cases, necessary and sufficient conditions are derived for two, three and four ordered matrices to be independent in the weighted Moore–Penrose inverse and the Drazin inverse. Some known results can be regarded as the special cases of the results in this paper.