The main objective of this paper is to introduce unique representations and characterizations for the weighted core inverse of matrices. We also investigate various properties of these inverses and their relationships with other generalized inverses. Proposed representations of the matrix-weighted core inverse will help us to discuss some results associated with the reverse order law for these inverses. Furthermore, this paper introduces an extension of the concepts of generalized bilateral inverse and $\{1,2,3,1^k\}$-inverse and their respective dual for complex rectangular matrices. Furthermore, we establish characterizations of EP-ness and the condition when both $W$-weighted $\{1,2,3\}$ and $W$-weighted $\{1,2,3,1^k\}$ inverses coincide. Then, a W-weighted index-MP, W-weighted MP-index, and W-weighted MP-index-MP matrices for rectangular complex matrices is introduced. In addition, we define the dual inverses for both weighted bilateral inverses and $\{1,2,3,1^k\}$-inverse. Characteristics that lead to self-duality in weighted bilateral inverses are also examined.

翻译：本文的主要目标是引入矩阵加权核逆的唯一表示与刻画。我们还研究了这些逆的各种性质及其与其他广义逆的关系。所提出的矩阵加权核逆的表示将有助于讨论这些逆的逆序律相关结果。此外，本文针对复矩形矩阵推广了广义双边逆与$\{1,2,3,1^k\}$-逆及其对偶的概念。进一步，我们建立了EP性的刻画，以及$W$-加权$\{1,2,3\}$-逆与$W$-加权$\{1,2,3,1^k\}$-逆一致的条件。然后，引入了复矩形矩阵的W-加权指数-MP、W-加权MP-指数与W-加权MP-指数-MP矩阵。同时，我们定义了加权双边逆与$\{1,2,3,1^k\}$-逆的对偶逆，并考察了加权双边逆中导致自对偶性的特征。