Analogues of the adjoint matrix for generalized inverses and corresponding Cramer rules

TL;DR

This paper develops determinantal representations of generalized inverses like the Moore-Penrose and Drazin inverses using analogues of the classical adjoint matrix, leading to Cramer rules for solving singular linear systems.

Contribution

Introduces new determinantal formulas for generalized inverses based on adjoint analogues, enabling explicit Cramer rules for solutions of singular systems.

Findings

Determinantal representations for Moore-Penrose and Drazin inverses.

Cramer rules for least squares and Drazin inverse solutions.

Expressions for products involving generalized inverses.

Abstract

In this article, we introduce determinantal representations of the Moore - Penrose inverse and the Drazin inverse which are based on analogues of the classical adjoint matrix. Using the obtained analogues of the adjoint matrix, we get Cramer rules for the least squares solution and for the Drazin inverse solution of singular linear systems. Finally, determinantal expressions for ${\rm {\bf A}}^{+} {\rm {\bf A}}$, ${\rm {\bf A}} {\rm {\bf A}}^{+}$, and ${\rm {\bf A}}^{D} {\rm {\bf A}}$ are presented.