In this paper, we introduce certain different characterizations and several new properties of the $ m $-weak group inverse of a complex matrix. Also, the relationship between the $ m $-weak group inverse and a nonsingular bordered matrix is established as well as the Cramer's rule for the solution of the restricted matrix equation that depends on the $ m $-weak group inverse.

Some new characterizations of the generalized Moore-Penrose inverse are proposed using range, null space, several matrix equations and projectors. Several representations of the generalized Moore-Penrose inverse are given. The relationships between the generalized Moore-Penrose inverse and other generalized inverses are discussed using core-EP decomposition. The generalized Moore-Penrose matrices are introduced and characterized. One relation between the generalized Moore-Penrose inverse and corresponding nonsingular border matrix is presented. In addition, applications of the generalized Moore-Penrose inverse in solving restricted matrix equations are studied.

In this paper, we discuss different characteristics of the BT-inverse of a square matrix introduced by Baksalary and Trenkler [On a generalized core inverse, Appl. Math. Comput., 236 (2014), 450-457]. While the BT-inverse is defined by a expression, we present some necessary and sufficient conditions for a matrix to be the BT-inverse. Then we give a canonical form of BT-inverse and investigate the relationships between BT-inverse and other generalized inverses by Core-EP decomposition. Some properties of BT-inverse concerned with some classes of special matrix are identified by Core-EP decomposition. Furthermore new representations of BT-inverse are given by the maximal classes of matrices.

In this paper, we give simple characterizations of the CMP inverse to be an EP matrix and simplify Theorem 2.10 in [New characterizations of the CMP inverse of matrices. Linear Multilinear Algebra. 2020;68(4):790–804]. Also, we characterize the CMP inverse of a matrix based on its range space and null space. Several different representations of the CMP inverse are given. In addition, some new characterizations of k-commutative equalities for some outer generalized inverse are investigated in terms of the CMP inverse.

In this paper, we revisit the core inverse introduced by Baksalary and Trenkler. We first give some new characterizations of the core inverse. Then, we give a new representation of the core inverse, which is related to A _T,S ⁽²⁾ .