A Note on Solutions of Linear Systems

We will consider Rohde's general form of -inverse of a matrix . The necessary and sufficient condition for consistency of a linear system will be represented. We will also be concerned with the minimal number of free parameters in Penrose's formula for obtaining the general solution of the linear system. These results will be applied for finding the general solution of various homogenous and nonhomogenous linear systems as well as for different types of matrix equations. 1. Introduction In this paper, we consider nonhomogeneous linear system in variables where is an matrix over the field of rank and is an matrix over . The set of all matrices over the complex field will be denoted by , . The set of all matrices over the complex field of rank will be denoted by . For simplicity of notation, we will write ( ) for the th row (the th column) of the matrix . Any matrix satisfying the equality is called -inverse of and is denoted by . The set of all -inverses of the matrix is denoted by . It can be shown that is not empty. If the matrix is invertible, then the equation has exactly one solution , so the only -inverse of the matrix is its inverse ; that is, = . Otherwise, -inverse of the matrix is not uniquely determined. For more information about -inverses and various generalized inverses, we recommend Ben-Israel and Greville [1] and Campbell and Meyer [2]. For each matrix there are regular matrices and such that where is identity matrix. It can be easily seen that every -inverse of the matrix can be represented in the form where , , and are arbitrary matrices of corresponding dimensions , , and with mutually independent entries; see Rohde [3] and Peri？ [4]. We will generalize the results of Urquhart [5]. Firstly, we explore the minimal numbers of free parameters in Penrose's formula for obtaining the general solution of the system (1). Then, we consider relations among the elements of to obtain the general solution in the form of the system (1) for . This construction has previously been used by Male？evi？ and Radi？i？ [6] (see also [7] and [8]). At the end of this paper, we will give an application of this results to the matrix equation . 2. The Main Result In this section, we indicate how a technique of an -inverse may be used to obtain the necessary and sufficient condition for an existence of a general solution of a nonhomogeneous linear system. Lemma 1. The nonhomogeneous linear system (1) has a solution if and only if the last coordinates of the vector are zeros, where is regular matrix such that (2) holds. Proof. The proof follows immediately from