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

Within the framework of the theory of row-column determinants previously introduced by the author, we get determinantal representations (analogs of Cramer ’ s rule) of a partial solution to the system of two-sided quaternion matrix equations A 1 XB 1 = C 1 , A 2 XB 2 = C 2 . We also give Cramer ’ s rules for its special cases when the first equation be one-sided. Namely, we consider the two systems with the first equation A 1 X = C 1 and XB 1 = C 1 , respec-tively, and with an unchanging second equation. Cramer ’ s rules for special cases when two equations are one-sided, namely the system of the equations A 1 X = C 1 , XB 2 = C 2 , and the system of the equations A 1 X = C 1 , A 2 X = C 2 are studied as well. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use its determinantal representa- tions previously obtained by the author in terms of row-column determinants as well.


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 (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.
Throughout the chapter, we denote the real number field by R, the set of all m Â n matrices over the quaternion algebra H ¼ a 0 þ a 1 i þ a 2 j þ a 3 k j i 2 ¼ j 2 ¼ k 2 ¼ À1; a 0 ; a 1 ; a 2 ; a 3 ∈ R È É by H mÂn and by H mÂn r , and the set of matrices over H with a rank r. For A ∈ H nÂm , the symbols A * stands for the conjugate transpose (Hermitian adjoint) matrix of A. The matrix A ¼ a ij À Á ∈ H nÂn is Hermitian if A * =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 mÂn , denoted by A † , is the unique matrix X ∈ H nÂm satisfying 1 ð ÞAXA ¼ A, 2 ð ÞXAX ¼ X, 3 ð Þ AX ð Þ * ¼ AX, and 4 ð Þ XA ð Þ * ¼ 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][10][11][12][13][14][15][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][20][21][22][23][24][25]) and by other reseachers (see, e.g. [26][27][28][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.

Preliminaries
For A ¼ a ij À Á ∈ M n; H ð Þ, we define n row determinants and n column determinants as follows. Suppose S n is the symmetric group on the set I n ¼ 1; …; n f g .
Definition 2.1. The ith row determinant of A ∈ H nÂm is defined for all i ¼ 1, …, n by putting with conditions i k2 < i k3 < … < i kr and i kt < i ktþs for all t ¼ 2, …, r and all s ¼ 1, …, l t .
Definition 2.2. The jth column determinant of A ∈ H nÂm is defined for all j ¼ 1, …, n by putting with conditions, j k2 < j k3 < … < j kr and j kt < j ktþs for t ¼ 2, …, r and s ¼ 1, …, l t .
Since  [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 consider the determinantal representations of the Moore-Penrose inverse.
We shall use the following notations. Let α≔ α 1 ; …; α k f g ⊆ 1; …; m f g and β≔ β 1 ; …; β k È É ⊆ 1; …; n f g be subsets of the order 1 ≤ k ≤ min m; n f g. A α β denotes the submatrix of A ∈ H nÂm determined by the rows indexed by α and the columns indexed by β. Then, A α α denotes the principal submatrix determined by the rows and columns indexed by α. If A ∈ H nÂn is Hermitian, then A j j α α is the corresponding principal minor of det A. For 1 ≤ k ≤ n, the collection of strictly increasing sequences of k integers chosen from 1; …; n f g is denoted by g . For fixed i ∈ α and j ∈ β, let I r, m i f g≔ α : α ∈ f L r, m ; i ∈ αg, J r, n j f g≔ β : β ∈ L r, n ; j ∈ β È É .
Let a :j be the jth column and a i: be the ith row of A. Suppose A :j b ð Þ denotes the matrix obtained from A by replacing its jth column with the column b, then A i: b ð Þ denotes the matrix obtained from A by replacing its ith row with the row b. a * :j and a * i: denote the jth column and the ith row of A * , respectively.
The following theorem gives determinantal representations of the Moore-Penrose inverse over the quaternion skew field H.
∈ H nÂm possesses the following determinantal representations: Remark 2.1. Note that for an arbitrary full-rank matrix, A ∈ H mÂn r , a column-vector d :j , and a rowvector d i: with appropriate sizes, respectively, we put and R A ≔I À AA † stand for some orthogonal projectors induced from A.
Theorem 2.2. [30] Let A ∈ H mÂn , B ∈ H rÂs , and C ∈ H mÂs be known and X ∈ H nÂr be unknown. Then, the matrix equation is consistent if and only if AA † CBB † ¼ C. In this case, its general solution can be expressed as where V and W are arbitrary matrices over H with appropriate dimensions.
∈ H nÂr has determinantal representations, are the column vector and the row vector, respectively.c i: andc :j are the ith row and the jth column of 3. Determinantal representations of a partial solution to the system (1)

Then, the system (1) is consistent if and only if
In that case, the general solution of (1) can be expressed as the following, where Z and W are the arbitrary matrices over H with compatible dimensions.
Cramer's Rules for the System of Two-Sided Matrix Equations and of Its Special Cases http://dx.doi.org/10.5772/intechopen.74105 Some simplification of (8) can be derived due to the quaternionic analog of the following proposition.

Lemma 3.2. [32]
If A ∈ H nÂn is Hermitian and idempotent, then the following equation holds for any matrix B ∈ H mÂn , It is evident that if A ∈ H nÂn is Hermitian and idempotent, then the following equation is true as well, Since L A1 , R B1 , and R H are projectors, then using (9) and (10), we have, respectively, Using (11) and (6), we obtain the following expression of (8), By putting Z 1 ¼ W 1 ¼ 0 in (12), the partial solution of (8) can be derived, Further we give determinantal representations of (13). Let Consider each term of (13) separately.
(i) By Theorem 2.3 for the first term, x 01 ij , of (13), we have or where are the column vector and the row vector, respectively.c 1 ð Þ q: andc 1 ð Þ :l are the qth row and the lth column ofC 1 (ii) Similarly, for the second term of (13), we have or (iii) The third term of (13) can be obtained by Theorem 2.3 as well. Then or are the column vector and the row vector, respectively. b c 2 ð Þ q: is the qth row and b c 2 ð Þ :l is the lth column of b C 2 ¼ T * C 2 N * . The following expression gives some simplify in computing. Since (iv) Using (3) for determinantal representations of H † and T † in the fourth term of (13), we obtain are the ith columns of the matrices H * A 2 and T * A 2 , respectively; q fj is the (fj)th element of Q B2 with the determinantal representation, (v) Similar to the previous case, (vi) Consider the sixth term by analogy to the fourth term. So, where or and are the column vector and the row vector, respectively. c q: 2 ð Þ and c :l 2 ð Þ are the qth row and the lth column of C 2 ¼ T * C 2 B * 2 for all i ¼ 1, …, n and j ¼ 1, …, p.
(vii) Using (3) for determinantal representations of and T † and (2) for N † in the seventh term of (13), we obtain where a 2;T ð Þ :q are the qth column of T * A 2 and the fth row of Hence, we prove the following theorem.

Cramer's rules for special cases of (1)
In this section, we consider special cases of (1) when one or two equations are one-sided. Let in Eq.(1), the matrix B 1 is vanished. Then, we have the system The following lemma is extended to matrices with quaternion entries.
[7] Let A 1 ∈ H mÂn , C 1 ∈ H mÂr , A 2 ∈ H kÂn , B 2 ∈ H rÂp , and C 2 ∈ H kÂp be given and X ∈ H nÂr is to be determined. Put H ¼ A 2 L A1 . Then, the following statements are equivalent: i.
In this case, the general solution of (27) can be expressed as where Z 1 and W 1 are the arbitrary matrices over H with appropriate sizes.
Since by (9), then we have some simplification of (28), By putting Z 1 =W 1 =0, there is the following partial solution of (27), Then, the partial solution (29), X 0 ¼ x 0 ij ∈ H nÂr , possesses the following determinantal representations, Cramer's Rules for the System of Two-Sided Matrix Equations and of Its Special Cases http://dx.doi.org/10.5772/intechopen.74105 In another case, and the column-vectors v The proof is similar to the proof of Theorem 3.1.
Let in Eq.(1), the matrix A 1 is vanished. Then, we have the system, The following lemma is extended to matrices with quaternion entries as well.
[7] Let B 1 ∈ H rÂs , C 1 ∈ H nÂs , A 2 ∈ H kÂn , B 2 ∈ H rÂp , and C 2 ∈ H kÂp be given and X ∈ H nÂr is to be determined. Put N ¼ R B1 B 2 . Then, the following statements are equivalent: In this case, the general solution of (30) can be expressed as where Z 2 and W 2 are the arbitrary matrices over H with appropriate sizes.
Since by (10), then some simplification of (31) can be derived, By putting Z 2 =W 2 =0, there is the following partial solution of (30), The following theorem on determinantal representations of (29) can be proven similar to the proof of Theorem 3.1 as well.
∈ H sÂr , and P A2 C 1 ≕P ¼p ij ∈ H nÂs . Then, the partial solution (32), X 0 ¼ x 0 ij ∈ H nÂr , possesses the following determinantal representations, [31] Let A 1 ∈ H mÂn , B 2 ∈ H rÂp , C 1 ∈ H mÂr , and C 2 ∈ H nÂp be given and X ∈ H nÂr is to be determined. Then, the system (33) is consistent if and only if R A1 C 1 ¼ 0, C 2 L B2 ¼ 0, and A 1 C 2 =C 1 B 2 . Under these conditions, the general solution to (33) can be established as where U is a free matrix over H with a suitable shape.
Due to the consistence conditions, Eq. (34) can be expressed as follows: Consequently, the partial solution X 0 to (33) is given by or Due to the expression (35), the following theorem can be proven similar to the proof of Theorem 3.1.
Then, the partial solution (35), X 0 ¼ x 0 ij ∈ H nÂs , possesses the following determinantal representation, Remark 4.1. In accordance to the expression (36), we obtain the same representations, but with the Let in Eq.(1), the matrices B 1 and B 2 are vanished. Then, we have the system Lemma 4.4. [7] Suppose that A 1 ∈ H mÂn , C 1 ∈ H mÂr , A 2 ∈ H kÂn , and C 2 ∈ H kÂr are known and X ∈ H nÂr is unknown,

Then, the system (38) is consistent if and only if
Under these conditions, the general solution to (38) can be established as where Y is an arbitrary matrix over H with an appropriate size.
Using (10) and the consistency conditions, we simplify (39) accordingly, Consequently, the following partial solution of (39) will be considered In the following theorem, we give the determinantal representations of (40).
H kÂr , and there exist A † 1 ¼ a ∈ H nÂs . Let rankH ¼ min rankA 2 ; rank f ∈ H nÂn . Then, X 0 ¼ x 0 ij ∈ H nÂr possesses the following determinantal representation, Proof. The proof is similar to the proof of Theorem 3.1.
Cramer's Rules for the System of Two-Sided Matrix Equations and of Its Special Cases http://dx.doi.org/10.5772/intechopen.74105

Conclusion
Within the framework of the theory of row-column determinants previously introduced by the author, we get determinantal representations (analogs of Cramer's rule) of partial solutions to the system of two-sided quaternion matrix equations A 1 XB 1 =C 1 , A 2 XB 2 =C 2 , and its special cases with 1 and 2 one-sided matrix equations. We use previously obtained by the author determinantal representations of the Moore-Penrose inverse. Note to give determinantal representations for all above matrix systems over the complex field, it is obviously needed to substitute all row and column determinants by usual determinants.