A Note on Solutions of Linear Systems

where A is anm× nmatrix over the field C of rank a and c is anm× 1matrix over C. The set of allm× nmatrices over the complex field C will be denoted by C, m, n ∈ N. The set of all m × n matrices over the complex field C of rank a will be denoted by C a . For simplicity of notation, we will write A i→ (A↓j) for the ith row (the jth column) of the matrixA ∈ C. Any matrix X satisfying the equality AXA = A is called {1}-inverse of A and is denoted by A. The set of all {1}inverses of the matrix A is denoted by A{1}. It can be shown that A{1} is not empty. If the n × n matrix A is invertible, then the equation AXA = A has exactly one solution A, so the only {1}-inverse of the matrix A is its inverse A; that is, A{1}={A}. Otherwise, {1}-inverse of the matrix A is not uniquely determined. For more information about {1}inverses and various generalized inverses, we recommend Ben-Israel and Greville [1] and Campbell and Meyer [2]. For each matrix A ∈ C a there are regular matrices P ∈ C and Q ∈ C such that


Introduction
In this paper, we consider nonhomogeneous linear system in variables where is an × matrix over the field C of rank and is an × 1 matrix over C. The set of all × matrices over the complex field C will be denoted by C × , , ∈ N. The set of all × matrices over the complex field C of rank will be denoted by C × . For simplicity of notation, we will write → ( ↓ ) for the th row (the th column) of the matrix ∈ C × .
Any matrix satisfying the equality = is called {1}-inverse of and is denoted by (1) . The set of all {1}inverses of the matrix is denoted by {1}. It can be shown that {1} is not empty. If the × matrix is invertible, then the equation = has exactly one solution −1 , so the only {1}-inverse of the matrix is its inverse −1 ; that is, {1}={ −1 }. Otherwise, {1}-inverse of the matrix is not uniquely determined. For more information about {1}inverses and various generalized inverses, we recommend Ben-Israel and Greville [1] and Campbell and Meyer [2].
For each matrix ∈ C × there are regular matrices ∈ C × and ∈ C × such that where is × identity matrix. It can be easily seen that every {1}-inverse of the matrix can be represented in the form  [3] and Perić [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 (1) to obtain the general solution in the form = (1) of the system (1) for ̸ = 0. This construction has previously been used by Malešević and Radičić [6] (see also [7] and [8]). At the end of this paper, we will give an application of this results to the matrix equation = .

The Main Result
In this section, we indicate how a technique of an {1}-inverse may be used to obtain the necessary and sufficient condition Proof. The proof follows immediately from Kronecker-Capelli theorem. We provide a new proof of the lemma by using the {1}-inverse of the system matrix . The system (1) has a solution if and only if = (1) ; see Penrose [9]. Since (1) is described by (3), it follows that Hence, we have the following equivalences: Furthermore, we conclude that = (1) ⇔ − = 0.

Theorem 2. The vector
∈ C ×1 is an arbitrary column, is the general solution Proof. Since {1}-inverse (1) of the matrix has the form (3), the solution of the system = (1) + ( − (1) ) can be represented in the form According to Lemma 1 and from (2), we have Furthermore, we obtain where = −1 . We now conclude that Therefore, since matrix is regular, we deduce that [ /( ( − ) + − )] is the general solution of the system (1) if and only if the rows of the matrix ( − )+ − are − free parameters.

Corollary 3. The vector
∈ C ×1 is an arbitrary column, is the general solution of the homogeneous linear system = 0, ∈ C × , if and only if the {1}-inverse (1) of the system matrix has the form (3) for arbitrary matrices and and the rows of the matrix − + Example 4. Consider the homogeneous linear system The system matrix is For regular matrices ISRN Algebra 3 equality (2) holds. Rohde's general {1}-inverse (1) of the system matrix is of the form According to Corollary 3 the general solution of the system (13) is of the form where Therefore, we obtain ] .
We are now concerned with the matrix equation where ∈ C × , ∈ C × , and ∈ C × .  Proof. Applying Theorem 2 on each system ↓ = ↓ , 1 ≤ ≤ , we obtain that From now on we proceed with the study of the nonhomogeneous linear system of the form where is an × matrix over the field C of rank and is a 1 × matrix over C. Let ∈ C × and let ∈ C × be regular matrices such that An {1}-inverse of the matrix can be represented in Rohde's form .

(35)
We can now proceed analogously to the proof of Theorem 2 to obtain that = [ is the general solution of the system = if and only if the rows of the matrix ( − ) + − are − free parameters. Therefore, is the general solution of the system (31) if and only if the columns of the matrix ( − ) + − are − free parameters. Analogous corollaries hold for Theorem 11.
We now deal with the matrix equation where ∈ C × , ∈ C × , and ∈ C × .

Lemma 12. Matrix equation (38) has a solution if and only if
the last − columns of the matrix = are zeros, where ∈ C × is regular matrix such that (32) holds.
Theorem 13. The matrix

An Application
In this section we will briefly sketch properties of the general solution of matrix equation where ∈ C × , ∈ C × , ∈ C × , and ∈ C × . If we denote by matrix product , then the matrix equation (41) becomes According to Theorem 9, the general solution of the system (42) can be presented as a product of the matrix and the matrix which has the first = rank( ) rows same as the matrix and the elements of the last − rows are ( − ) mutually independent free parameters; and are regular matrices such that = . Thus, we are now turning on to the system of the form = . (43) By Theorem 13, we conclude that the general solution of the system (43) can be presented as a product of the matrix which has the first = rank( ) columns equal to the first columns of the matrix and the rest of the columns have mutually independent free parameters as entries, and the matrix , for regular matrices and such that = . Therefore, the general solution of the system (41) is of the form where is a submatrix of the matrix corresponding to the first rows and the first columns and the entries of the matrices , , and are − free parameters. We will illustrate this on the following example.
Example 14. We consider the matrix equation for an arbitrary matrix = [ 11 12 13 21 − 22 23 ]. Therefore, the general solution of the system = is From now on, we consider the system for an arbitrary matrix = [ 11 12 13 21 22 23 ]. Finally, the solution of the system = is