  This paper introduces new classes of generalized inverses for square matrices
named GD1, and the dual, called 1GD inverse. In addition, we discuss a few
characterizations and representations of these inverses. The explicit
expressions of these inverses have been established via core-nilpotent
decomposition. Further, we introduce a binary relation for GD1 inverse and 1GD
inverse, along with a few derived properties.
