A note on solutions of linear systems

TL;DR

This paper explores Rohde's generalized inverse of a matrix, providing conditions for system consistency and analyzing the minimal free parameters in Penrose's formula to find general solutions of linear systems.

Contribution

It presents a detailed analysis of Rohde's {1}-inverse, conditions for system consistency, and the minimal free parameters in Penrose's formula for solving linear systems.

Findings

Condition for system consistency derived

Minimal free parameters in Penrose's formula identified

Applications to various linear and matrix equations

Abstract

In this paper we will consider Rohde's general form of {1}-inverse of a matrix A. The necessary and sufficient condition for consistency of a linear system Ax=c will be represented. We will also be concerned with the minimal number of free parameters in Penrose's formula x = A^(1) c + (I - A^(1)A)y for obtaining the general solution of the linear system. This results will be applied for finding the general solution of various homogenous and non-homogenous linear systems as well as for different types of matrix equations.