A note on solutions of linear systems

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^1A)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.