1. Introduction

The study of matrix equations and systems of matrix equations is an active research topic in matrix theory and its applications. The system of classical two-sided matrix equations

$$\begin{cases} \mathbf{A}\_1 \mathbf{X} \mathbf{B}\_1 = \mathbf{C}\_1 \\ \mathbf{A}\_2 \mathbf{X} \mathbf{B}\_2 = \mathbf{C}\_2. \end{cases} \tag{1}$$

over the complex field, a principle domain, and the quaternion skew field has been studied by many authors (see, e.g. [1–7]). Mitra [1] gives necessary and sufficient conditions of the system

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

(1) over the complex field and the expression for its general solution. Navarra et al. [6] derived a new necessary and sufficient condition for the existence and a new representation of (1) over the complex field and used the results to give a simple representation. Wang [7] considers the system (1) over the quaternion skew field and gets its solvability conditions and a representation of a general solution.

2. Preliminaries

with conditions, j

Hermitian, then j j <sup>A</sup> <sup>α</sup>

For <sup>A</sup> <sup>¼</sup> aij � �<sup>∈</sup> <sup>M</sup>ð Þ <sup>n</sup>; <sup>H</sup> , we define <sup>n</sup> row determinants and <sup>n</sup> column determinants as follows.

aik<sup>1</sup> ik1þ<sup>1</sup>…aik1þ<sup>l</sup>

σ ¼ iik<sup>1</sup> ik1þ<sup>1</sup>…ik1þl<sup>1</sup> ð Þ ik<sup>2</sup> ik2þ<sup>1</sup>…ik2þl<sup>2</sup> ð Þ… ikr ikrþ<sup>1</sup>…ikrþlr ð Þ,

Definition 2.2. The jth column determinant of <sup>A</sup> <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>m</sup> is defined for all j <sup>¼</sup> <sup>1</sup>, …, n by putting

k2þl<sup>2</sup> …j <sup>k</sup>2þ<sup>1</sup><sup>j</sup> k2

Since rdet1<sup>A</sup> <sup>¼</sup> <sup>⋯</sup> <sup>¼</sup> rdetn<sup>A</sup> <sup>¼</sup> cdet1<sup>A</sup> <sup>¼</sup> <sup>⋯</sup> <sup>¼</sup> cdetn<sup>A</sup> <sup>∈</sup> <sup>R</sup> for Hermitian <sup>A</sup> <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>n</sup>, then we can define the determinant of a Hermitian matrix A by putting, detA≔rdet<sup>i</sup> A ¼ cdetiA, for all i ¼ 1, …, n. The determinant of a Hermitian matrix has properties similar to a usual determinant. They are completely explored in [17, 18] by its row and column determinants. In particular, within the framework of the theory of the column-row determinants, the determinantal representations of the inverse matrix over H by analogs of the classical adjoint matrix and Cramer's rule for quaternionic systems of linear equations have been derived. Further, we

We shall use the following notations. Let α≔f g α1;…; α<sup>k</sup> ⊆f g 1;…; m and β≔ β1;…; β<sup>k</sup>

principal submatrix determined by the rows and columns indexed by α. If A ∈ H<sup>n</sup>�<sup>n</sup> is

of strictly increasing sequences of k integers chosen from 1f g ;…; n is denoted by Lk,n≔ α : α ¼ ð Þ α1; …; α<sup>k</sup> f g ; 1 ≤ α<sup>1</sup> ≤…≤ α<sup>k</sup> ≤ n . For fixed i∈ α and j∈ β, let Ir,mf gi ≔fα : α ∈

Let a:<sup>j</sup> be the jth column and a<sup>i</sup>: be the ith row of A. Suppose A:<sup>j</sup>ð Þ b denotes the matrix obtained from A by replacing its jth column with the column b, then A<sup>i</sup>:ð Þ b denotes the matrix obtained

<sup>α</sup> is the corresponding principal minor of det A. For 1 ≤ k ≤ n, the collection

determined by the rows indexed by α and the columns indexed by β. Then, A<sup>α</sup>

kt < j

� � <sup>j</sup>

1 i � �… aikr ikrþ<sup>1</sup>…aikrþlr ikr

Cramer's Rules for the System of Two-Sided Matrix Equations and of Its Special Cases

� �,

http://dx.doi.org/10.5772/intechopen.74105

5

1 …aj <sup>k</sup>1þ1<sup>j</sup> k1 aj k1 j

k1þl<sup>1</sup> …j <sup>k</sup>1þ<sup>1</sup><sup>j</sup> k1 j

ktþ<sup>s</sup> for <sup>t</sup> <sup>¼</sup> <sup>2</sup>, …, r and <sup>s</sup> <sup>¼</sup> <sup>1</sup>,…, lt.

� �,

� �⊆

<sup>α</sup> denotes the

<sup>β</sup> denotes the submatrix of A ∈ H<sup>n</sup>�<sup>m</sup>

� �,

Definition 2.1. The ith row determinant of <sup>A</sup> <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>m</sup> is defined for all i <sup>¼</sup> <sup>1</sup>, …, n by putting

with conditions ik<sup>2</sup> < ik<sup>3</sup> < … < ikr and ikt < iktþ<sup>s</sup> for all t ¼ 2, …, r and all s ¼ 1, …, lt.

kr j krþlr …aj krþ<sup>1</sup>ikr � �… aj jk1þ<sup>l</sup>

ð Þ �<sup>1</sup> <sup>n</sup>�<sup>r</sup> aiik<sup>1</sup>

ð Þ �<sup>1</sup> <sup>n</sup>�<sup>r</sup> aj

kr and j

consider the determinantal representations of the Moore-Penrose inverse.

Suppose Sn is the symmetric group on the set In ¼ f g 1;…; n .

σ ∈Sn

rdeti<sup>A</sup> <sup>¼</sup> <sup>X</sup>

cdet<sup>j</sup> <sup>A</sup> <sup>¼</sup> <sup>X</sup>

τ ¼ j

<sup>k</sup><sup>2</sup> < j

Lr,m; <sup>i</sup>∈αg, Jr,nf g<sup>j</sup> <sup>≔</sup> <sup>β</sup> : <sup>β</sup> <sup>∈</sup>Lr,n; <sup>j</sup>∈<sup>β</sup> � �.

τ∈ Sn

krþlr …j krþ<sup>1</sup><sup>j</sup> kr � �… <sup>j</sup>

<sup>k</sup><sup>3</sup> < … < j

f g <sup>1</sup>;…; <sup>n</sup> be subsets of the order 1 <sup>≤</sup> <sup>k</sup> <sup>≤</sup> minf g <sup>m</sup>; <sup>n</sup> . <sup>A</sup><sup>α</sup>

Throughout the chapter, we denote the real number field by R, the set of all m � n matrices over the quaternion algebra

$$\mathbb{H} = \left\{ a\_0 + a\_1 \mathbf{i} + a\_2 \mathbf{j} + a\_3 \mathbf{k} \, | \, \mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = -1, a\_0, a\_1, a\_2, a\_3 \in \mathbb{R} \right\}$$

by H<sup>m</sup>�<sup>n</sup> and by H<sup>m</sup>�<sup>n</sup> <sup>r</sup> , and the set of matrices over H with a rank r. For A ∈ H<sup>n</sup>�<sup>m</sup>, the symbols A\* stands for the conjugate transpose (Hermitian adjoint) matrix of A. The matrix <sup>A</sup> <sup>¼</sup> aij <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>n</sup> is Hermitian if <sup>A</sup>\* =A.

Generalized inverses are useful tools used to solve matrix equations. The definitions of the Moore-Penrose inverse matrix have been extended to quaternion matrices as follows. The Moore-Penrose inverse of A ∈ H<sup>m</sup>�<sup>n</sup>, denoted by A† , is the unique matrix X∈ H<sup>n</sup>�<sup>m</sup> satisfying ð Þ<sup>1</sup> AXA <sup>¼</sup> <sup>A</sup>, 2ð ÞXAX <sup>¼</sup> <sup>X</sup>, 3ð Þð Þ AX <sup>∗</sup> <sup>¼</sup> AX, and 4ð Þð Þ XA <sup>∗</sup> <sup>¼</sup> XA.

The determinantal representation of the usual inverse is the matrix with the cofactors in the entries which suggests a direct method of finding of inverse and makes it applicable through Cramer's rule to systems of linear equations. The same is desirable for the generalized inverses. But there is not so unambiguous even for complex or real generalized inverses. Therefore, there are various determinantal representations of generalized inverses because of looking for their more applicable explicit expressions (see, e.g. [8]). Through the noncommutativity of the quaternion algebra, difficulties arise already in determining the quaternion determinant (see, e.g. [9–16]).

The understanding of the problem for determinantal representation of an inverse matrix as well as generalized inverses only now begins to be decided due to the theory of column-row determinants introduced in [17, 18]. Within the framework of the theory of column-row determinants, determinantal representations of various kinds of generalized inverses and (generalized inverses) solutions of quaternion matrix equations have been derived by the author (see, e.g. [19–25]) and by other reseachers (see, e.g. [26–29]).

The main goals of the chapter are deriving determinantal representations (analogs of the classical Cramer rule) of general solutions of the system (1) and its simpler cases over the quaternion skew field.

The chapter is organized as follows. In Section 2, we start with preliminaries introducing of row-column determinants and determinantal representations of the Moore-Penrose and Cramer's rule of the quaternion matrix equations, AXB=C. Determinantal representations of a partial solution (an analog of Cramer's rule) of the system (1) are derived in Section 3. In Section 4, we give Cramer's rules to special cases of (1) with 1 and 2 one-sided equations. Finally, the conclusion is drawn in Section 5.
