The equalities (Formula presented.) and (Formula presented.) are called the reverse order law and forward order law, respectively, for invertible matrices. However, these generally fail for singular/rectangular matrices, tensors of arbitrary order, and operators. Several researchers have worked in these problems starting with Greville [SIAM Rev. 8 (1966), 518–521: MR0210720]. Recently, Višnjić et al. [Electron. J. Linear Algebra 39 (2023), 379–394: MR4626656] studied the reverse order law for the (Formula presented.) -inverse in a unital ring. This manuscript establishes several sufficient conditions for the validity of both the reverse order law and forward order law for the new dual Moore–Penrose inverse (NDMPI) introduced in [J. Comput. Appl. Math., 454 (2025), 116185: MR4782122]. Additionally, some characterizations of the reverse order law of the NDMPI are obtained. We also explore a few possible theoretical applications of the reverse order law within this framework. Finally, we illustrate the additive property for dual complex matrices.

Motivated by the very recent work of Gao, Y., Chen, J., Wang, J., Zou, H. [Comm. Algebra, 49(8) (2021) 3241-3254; MR4283143], we introduce two new generalized inverses named weak Drazin (WD) and weak Drazin Moore-Penrose (WDMP) inverses for elements in rings. A few of their properties are then provided, and the fact that the proposed generalized inverses coincide with different well-known generalized inverses under certain assumptions is established. Further, we discuss additive properties, reverse-order law and forward-order law for WD and WDMP generalized inverses. Then, we propose a binary relation called the WD order. Some examples are also provided in support of the theoretical results.

Motivated by the works of Wang and Liu (Linear Algebra Appl 488:235–248, 2016) and Mosić (Results Math 75(2):1–21, 2020), we provide further results on GD inverses, and then 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 exploit their various properties and characterize them in terms of various generalized inverses. We present a representation of a GD-star matrix by using the core-EP decomposition and Hartwig–Spindelböck 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.

This article establishes a few sufficient conditions of the forward-order law for the core inverse of elements in rings with involution. It also presents the forward-order law for the weighted core inverse and the triple forward-order law for the core inverse. Additionally, we discuss the hybrid forward-order law involving different generalized inverses like the Moore–Penrose inverse, the group inverse, and the core inverse.

In this article, we introduce two new generalized inverses for rectangular matrices called W-weighted generalized-Drazin–Moore–Penrose (GDMP) and W-weighted generalized-Drazin-reflexive (GDR) inverses. The first generalized inverse can be seen as a generalization of the recently introduced GDMP inverse for a square matrix to a rectangular matrix. The second class of generalized inverse contains the class of the first generalized inverse. We then exploit their various properties and establish that the proposed generalized inverses coincide with different well-known generalized inverses under certain assumptions. We also obtain a representation of W-weighted GDMP inverse employing EP-core nilpotent decomposition. We define the dual of W-weighted GDMP inverse and obtain analogue results. Further, we discuss additive properties, reverse-and forward-order laws for GD, W-weighted GD, GDMP, and W-weighted GDMP generalized inverses.