  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.
