An analog of the adjugate matrix for the outer inverse \(A^{(2)}_{T, S}\) (Q1954958)

Summary: We investigate the determinantal representation by exploiting the limiting expression for the generalized inverse \(A^{(2)}_{T, S}\). We show the equivalent relationship between the existence and limiting expression of \(A^{(2)}_{T, S}\) and some limiting processes of matrices and deduce the new determinantal representations of \(A^{(2)}_{T, S}\), based on some analog of the classical adjoint matrix. Using the analog of the classical adjoint matrix, we present Cramer rules for the restricted matrix equation \(AXB = D, \mathcal R(X) \subset T, \mathcal N(X) \supset \tilde{S}\).