  Motivated by the works of Wang and Liu [Linear Algebra Appl., 488 (2016)
235-248; MR3419784] and Mosic [Results Math., 75(2) (2020) 1-21; MR4079761], we
provide further results on GD inverses and introduce two new classes for square
matrices called GD-star (generalized-Drazin-star) and GD-star-one
(generalized-Drazin-star-one) using a GD inverse of a matrix. We then exploit
their various properties and characterize them in terms of various generalized
inverses. We establish a representation of a GD-star matrix by using the
core-nilpotent decomposition and Hartwig-Spindelbock decomposition. We also
define a binary relation called GD-star order using this class of matrices.
Further, we obtain some analogous results for the class of star-GD matrices.
Moreover, the reverse-order law and forward-order law for GD inverse along with
its monotonicity criteria are obtained.
