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 and representations 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. 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。我们进一步研究了这些逆的各种性质与表示，以及它们与其他广义逆之间的关系。所提出的矩阵加权核逆表示将有助于讨论关于这些逆的反序律相关结果。此外，本文引入了广义双边逆、$\{1,2,3,1^k\}$-逆及其在复矩形矩阵上的对偶概念的推广。进一步，我们建立了EP性以及$W$-加权$\{1,2,3\}$逆与$W$-加权$\{1,2,3,1^k\}$逆一致的条件刻画。另外，我们定义了加权双边逆与$\{1,2,3,1^k\}$-逆的对偶逆。同时，还探讨了导致加权双边逆自对偶性的特征。