Determinantal representation of weighted generalized inverses

In this article, we introduce new determinantal representations of the weighted generalized inverse by using a lemma of Robinson [D.W. Robinson, The classical adjoint, Linear Algebra Appl. 411 (2005) 254–276]. And, using generalized Cramer’s rule, we get the method for computing minors of the weighted Moore–Penrose inverse and the W-weighted Drazin inverse.