The Mixed-type Reverse-order Laws For Some Generalized Inverses Of The Three-matrix Product

Suppose that the product ABC of three nonsingular matrices A, B, C exists, then the inverse of the matrix product satisfies the following reverse order law: (ABC)-1=C-1B-1A-1. The inverse can also be expressed in the form of mixed-type reverse order law: (ABC)-1=(BC)-1B(AB)-1. However, this so-called mixed-type reverse order law is not necessarily true for gen-eralized inverses of the matrices if the product ABC is singular. Therefore, one of the fundamental research problems in the theory of generalized inverses of matrices is how to find the necessary and sufficient conditions for the mixed-type reverse or-der laws for generalized inverses of matrices product to hold. Since the 1960s, many researchers have paid attention to them and some interesting results have been ob-tained. Generally, the mixed-type reverse-order law for{i,j,k}-inverse of a triple matrix product ABC can be written as (ABC)(i,j,k)＝(BC)(i,j,k)B(AB)(i,j,k). However, such mixed-type reverse-order law for{i,j,k}-inverse is generally not true and so needs to investigate the conditions for it holding. Y.Tian ([4]) have studied several special cases.In this paper, we will continuously focus on this topic and obtain the necessary and sufficient conditions for the following mixed-type set inclusions {(BC)(1,i,j)B(AB){1,i,j)}(?){(ABC){1,i,j)},i=2,3; j=3,4 holding by using the ranks of matrices. Some related results are also been obtained.