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

Abstract

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 A1XB1=C1, A2XB2=C2. 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 A1X=C1 and XB1=C1, respectively, and with an unchanging second equation. Cramer’s rules for special cases when two equations are one-sided, namely the system of the equations A1X=C1, XB2=C2, and the system of the equations A1X=C1, A2X=C2 are studied as well. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use its determinantal representations previously obtained by the author in terms of row-column determinants as well.

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

$$\left\{\begin{array}{l}{\mathbf{A}}_1{\mathbf{XB}}_1={\mathbf{C}}_1,\\ {\mathbf{A}}_2{\mathbf{XB}}_2={\mathbf{C}}_2.\end{array}\right.$$ E1

over the complex field, a principle domain, and the quaternion skew field has been studied by many authors (see, e.g. [1, 2, 3, 4, 5, 6, 7]). Mitra [1] gives necessary and sufficient conditions of the system (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 $\mathbb{R}$ , the set of all $m\times n$ matrices over the quaternion algebra

$$\mathbb{H}=\left\{a_0+a_{1}i+a_{2}j+a_{3}k|i^2=j^2=k^2=-1,a_0,a_1,a_2,a_3\in \mathbb{R}\right\}$$

by ${\mathbb{H}}^{m\times n}$ and by ${\mathbb{H}}_r^{m\times n}$ , and the set of matrices over $\mathbb{H}$ with a rank r. For $\mathbf{A}\in {\mathbb{H}}^{n\times m}$ , the symbols A^* stands for the conjugate transpose (Hermitian adjoint) matrix of A. The matrix $\mathbf{A}=\left(a_{\mathit{ij}}\right)\in {\mathbb{H}}^{n\times n}$ 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 $\mathbf{A}\in {\mathbb{H}}^{m\times n}$ , denoted by ${\mathbf{A}}^{†}$ , is the unique matrix $\mathbf{X}\in {\mathbb{H}}^{n\times m}$ satisfying $\left(1\right)\mathbf{AXA}=\mathbf{A}$ , $\left(2\right)\mathbf{XAX}=\mathbf{X}$ , $\left(3\right){\left(\mathbf{AX}\right)}^{\ast }=\mathbf{AX}$ , and $\left(4\right){\left(\mathbf{XA}\right)}^{\ast }=\mathbf{XA}$ .

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.

2. Preliminaries

For $\mathbf{A}=\left(a_{\mathit{ij}}\right)\in \mathrm{M}\left(n,\mathbb{H}\right)$ , we define n row determinants and n column determinants as follows. Suppose S_n is the symmetric group on the set $I_n=\left\{1,\dots ,n\right\}$ .

Definition 2.1. The ith row determinant of $\mathbf{A}\in {\mathbb{H}}^{n\times m}$ is defined for all $i=1,\dots ,n$ by putting

$${\text{rdet}}_i\mathbf{A}=\sum _{\sigma \in S_n}{\left(-1\right)}^{n-r}\left(a_{{\mathit{ii}}_{k_1}}{a}_{i_{k_1}{i}_{k_1+1}}\dots a_{i_{k_1+l_1}i}\right)\dots \left(a_{i_{k_r}{i}_{k_r+1}}\dots a_{i_{k_r+l_r}{i}_{k_r}}\right),$$

$$\sigma =\left(i{i}_{k_1}{i}_{k_1+1}\dots i_{k_1+l_1}\right)\left(i_{k_2}{i}_{k_2+1}\dots i_{k_2+l_2}\right)\dots \left(i_{k_r}{i}_{k_r+1}\dots i_{k_r+l_r}\right),$$

with conditions $i_{k_2}<i_{k_3}<\dots <i_{k_r}$ and $i_{k_t}<i_{k_t+s}$ for all $t=2,\dots ,r$ and all $s=1,\dots ,l_t$ .

Definition 2.2. The jth column determinant of $\mathbf{A}\in {\mathbb{H}}^{n\times m}$ is defined for all $j=1,\dots ,n$ by putting

$${\text{cdet}}_j\mathbf{A}=\sum _{\tau \in S_n}{\left(-1\right)}^{n-r}\left(a_{j_{k_r}{j}_{k_r+l_r}}\dots a_{j_{k_r+1}{i}_{k_r}}\right)\dots \left(a_{j{j}_{k_1+l_1}}\dots a_{j_{k_1+1}{j}_{k_1}}{a}_{j_{k_1}j}\right),$$

$$\tau =\left(j_{k_r+l_r}\dots j_{k_r+1}{j}_{k_r}\right)\dots \left(j_{k_2+l_2}\dots j_{k_2+1}{j}_{k_2}\right)\left(j_{k_1+l_1}\dots j_{k_1+1}{j}_{k_1}j\right),$$

with conditions, $j_{k_2}<j_{k_3}<\dots <j_{k_r}$ and $j_{k_t}<j_{k_t+s}$ for $t=2,\dots ,r$ and $s=1,\dots ,l_t$ .

Since ${\text{rdet}}_1\mathbf{A}=\cdots ={\text{rdet}}_n\mathbf{A}={\text{cdet}}_1\mathbf{A}=\cdots ={\text{cdet}}_n\mathbf{A}\in \mathbb{R}$ for Hermitian $\mathbf{A}\in {\mathbb{H}}^{n\times n}$ , then we can define the determinant of a Hermitian matrix A by putting, $\text{det}\mathbf{A}≔{\text{rdet}}_i\mathbf{A}={\text{cdet}}_i\mathbf{A}$ , for all $i=1,\dots ,n$ . 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 $\mathbb{H}$ 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 $\alpha ≔\left\{{\alpha }_1,\dots ,{\alpha }_k\right\}\subseteq \left\{1,\dots ,m\right\}$ and $\beta ≔\left\{{\beta }_1,\dots ,{\beta }_k\right\}\subseteq \left\{1,\dots ,n\right\}$ be subsets of the order $1\le k\le \text{min}\left\{m,n\right\}$ . ${\mathbf{A}}_{\beta }^{\alpha }$ denotes the submatrix of $\mathbf{A}\in {\mathbb{H}}^{n\times m}$ determined by the rows indexed by α and the columns indexed by β. Then, ${\mathbf{A}}_{\alpha }^{\alpha }$ denotes the principal submatrix determined by the rows and columns indexed by α. If $\mathbf{A}\in {\mathbb{H}}^{n\times n}$ is Hermitian, then ${\left|\mathbf{A}\right|}_{\alpha }^{\alpha }$ is the corresponding principal minor of det A. For $1\le k\le n$ , the collection of strictly increasing sequences of k integers chosen from $\left\{1,\dots ,n\right\}$ is denoted by $L_{k,n}≔\left\{\alpha :\alpha =\left({\alpha }_1,\dots ,{\alpha }_k\right),1\le {\alpha }_1\le \dots \le {\alpha }_k\le n\right\}$ . For fixed $i\in \alpha$ and $j\in \beta$ , let $I_{r,m}\left\{i\right\}≔\left\{\alpha :\alpha \in L_{r,m},i\in \alpha \right\},J_{r,n}\left\{j\right\}≔\left\{\beta :\beta \in L_{r,n},j\in \beta \right\}$ .

Let ${\mathbf{a}}_{.j}$ be the jth column and ${\mathbf{a}}_{i.}$ be the ith row of A. Suppose ${\mathbf{A}}_{.j}\left(\mathbf{b}\right)$ denotes the matrix obtained from A by replacing its jth column with the column b, then ${\mathbf{A}}_{i.}\left(\mathbf{b}\right)$ denotes the matrix obtained from A by replacing its ith row with the row b. ${\mathbf{a}}_{.j}^{\ast }$ and ${\mathbf{a}}_{i.}^{\ast }$ 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 $\mathbb{H}$ .

Theorem 2.1. [19] If $\mathbf{A}\in {\mathbb{H}}_r^{m\times n}$ , then the Moore-Penrose inverse ${\mathbf{A}}^{†}=\left(a_{\mathit{ij}}^{†}\right)\in {\mathbb{H}}^{n\times m}$ possesses the following determinantal representations:

$$a_{\mathit{ij}}^{†}=\frac{\sum _{\beta \in J_{r,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}^{\ast }\mathbf{A}\right)}_{.i}\left({\mathbf{a}}_{.j}^{\ast }\right)\right)}_{\beta }^{\beta }}{\sum _{\beta \in J_{r,n}}{\left|{\mathbf{A}}^{\ast }\mathbf{A}\right|}_{\beta }^{\beta }},$$ E2

or

$$a_{\mathit{ij}}^{†}=\frac{\sum _{\alpha \in I_{r,m}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{AA}}^{\ast }\right)}_{j.}\left({\mathbf{a}}_{i.}^{\ast }\right)\right)}_{\alpha }^{\alpha }}{\sum _{\alpha \in I_{r,m}}{\left|{\mathbf{AA}}^{\ast }\right|}_{\alpha }^{\alpha }}.$$ E3

Remark 2.1. Note that for an arbitrary full-rank matrix, $\mathbf{A}\in {\mathbb{H}}_r^{m\times n}$ , a column-vector ${\mathbf{d}}_{.j}$ , and a row-vector ${\mathbf{d}}_{i.}$ with appropriate sizes, respectively, we put

$${\text{cdet}}_i\left({\left({\mathbf{A}}^{\ast }\mathbf{A}\right)}_{.i}\left({\mathbf{d}}_{.j}\right)\right)=\sum _{\beta \in J_{n,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}^{\ast }\mathbf{A}\right)}_{.i}\left({\mathbf{d}}_{.j}\right)\right)}_{\beta }^{\beta },\text{det}\left({\mathbf{A}}^{\ast }\mathbf{A}\right)=\sum _{\beta \in J_{n,n}}{\left|{\mathbf{A}}^{\ast }\mathbf{A}\right|}_{\beta }^{\beta }\text{when}r=n,$$

$${\text{rdet}}_j\left({\left({\mathbf{AA}}^{\ast }\right)}_{j.}\left({\mathbf{d}}_{i.}\right)\right)=\sum _{\alpha \in I_{m,m}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{AA}}^{\ast }\right)}_{j.}\left({\mathbf{d}}_{i.}\right)\right)}_{\alpha }^{\alpha },\text{det}\left({\mathbf{AA}}^{\ast }\right)=\sum _{\alpha \in I_{m,m}}{\left|{\mathbf{AA}}^{\ast }\right|}_{\alpha }^{\alpha }\text{when}r=m.$$

Furthermore, ${\mathbf{P}}_A={\mathbf{A}}^{†}\mathbf{A}$ , ${\mathbf{Q}}_A={\mathbf{AA}}^{†}$ , ${\mathbf{L}}_A=\mathbf{I}-{\mathbf{A}}^{†}\mathbf{A}$ , and ${\mathbf{R}}_A≔\mathbf{I}-{\mathbf{AA}}^{†}$ stand for some orthogonal projectors induced from A.

Theorem 2.2. [30] Let $\mathbf{A}\in {\mathbb{H}}^{m\times n}$ , $\mathbf{B}\in {\mathbb{H}}^{r\times s}$ , and $\mathbf{C}\in {\mathbb{H}}^{m\times s}$ be known and $\mathbf{X}\in {\mathbb{H}}^{n\times r}$ be unknown. Then, the matrix equation

$$\mathbf{AXB}=\mathbf{C}$$ E4

is consistent if and only if ${\mathbf{AA}}^{†}{\mathbf{CBB}}^{†}=\mathbf{C}$ . In this case, its general solution can be expressed as

$$\mathbf{X}={\mathbf{A}}^{†}{\mathbf{CB}}^{†}+{\mathbf{L}}_A\mathbf{V}+{\mathbf{WR}}_B,$$ E5

where V and W are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions.

The partial solution, ${\mathbf{X}}^0={\mathbf{A}}^{†}{\mathbf{CB}}^{†}$ , of (4) possesses the following determinantal representations.

Theorem 2.3. [20] Let $\mathbf{A}\in {\mathbb{H}}_{r_1}^{m\times n}$ and $\mathbf{B}\in {\mathbb{H}}_{r_2}^{r\times s}$ . Then, ${\mathbf{X}}^0=\left(x_{\mathit{ij}}^0\right)\in {\mathbb{H}}^{n\times r}$ has determinantal representations,

$$x_{\mathit{ij}}=\frac{\sum _{\beta \in J_{r_1,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}^{\ast }\mathbf{A}\right)}_{.i}\left({\mathbf{d}}_{.j}^B\right)\right)}_{\beta }^{\beta }}{\sum _{\beta \in J_{r_1,n}}{\left|{\mathbf{A}}^{\ast }\mathbf{A}\right|}_{\beta }^{\beta }\sum _{\alpha \in I_{r_2,r}}{\left|{\mathbf{BB}}^{\ast }\right|}_{\alpha }^{\alpha }},$$

or

$$x_{\mathit{ij}}=\frac{\sum _{\alpha \in I_{r_2,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{BB}}^{\ast }\right)}_{j.}\left({\mathbf{d}}_{i.}^A\right)\right)}_{\alpha }^{\alpha }}{\sum _{\beta \in J_{r_1,n}}{\left|{\mathbf{A}}^{\ast }\mathbf{A}\right|}_{\beta }^{\beta }\sum _{\alpha \in I_{r_2,r}}{\left|{\mathbf{BB}}^{\ast }\right|}_{\alpha }^{\alpha }},$$

where

$${\mathbf{d}}_{.j}^B=\left[\sum _{\alpha \in I_{r_2,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{BB}}^{\ast }\right)}_{j.}\left({\tilde{\mathbf{c}}}_{k.}\right)\right)}_{\alpha }^{\alpha }\right]\in {\mathbb{H}}^{n\times 1},k=1,\dots ,n,$$

$${\mathbf{d}}_{i.}^A=\left[\sum _{\beta \in J_{r_1,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}^{\ast }\mathbf{A}\right)}_{.i}\left({\tilde{\mathbf{C}}}_{.l}\right)\right)}_{\beta }^{\beta }\right]\in {\mathbb{H}}^{1\times r},l=1,\dots ,r,$$

are the column vector and the row vector, respectively. ${\tilde{\mathbf{c}}}_{i.}$ and ${\tilde{\mathbf{c}}}_{.j}$ are the ith row and the jth column of $\tilde{\mathbf{C}}={\mathbf{A}}^{\ast }{\mathbf{CB}}^{\ast }$ .

3. Determinantal representations of a partial solution to the system (1)

Lemma 3.1. [7] Let ${\mathbf{A}}_1\in {\mathbb{H}}^{m\times n}$ , ${\mathbf{B}}_1\in {\mathbb{H}}^{r\times s}$ , ${\mathbf{C}}_1\in {\mathbb{H}}^{m\times s}$ , ${\mathbf{A}}_2\in {\mathbb{H}}^{k\times n}$ , ${\mathbf{B}}_2\in {\mathbb{H}}^{r\times p}$ , and ${\mathbf{C}}_2\in {\mathbb{H}}^{k\times p}$ be given and $\mathbf{X}\in {\mathbb{H}}^{n\times r}$ is to be determined. Put $\mathbf{H}={\mathbf{A}}_2{\mathbf{L}}_{A_1}$ , $\mathbf{N}={\mathbf{R}}_{B_1}{\mathbf{B}}_2$ , $\mathbf{T}={\mathbf{R}}_H{\mathbf{A}}_2$ , and $\mathbf{F}={\mathbf{B}}_2{\mathbf{L}}_N$ . Then, the system (1) is consistent if and only if

$${\mathbf{A}}_i{\mathbf{A}}_i^{†}{\mathbf{C}}_i{\mathbf{B}}_i^{†}{\mathbf{B}}_i={\mathbf{C}}_i,i=1,2;$$ E6

$$\mathbf{T}\left[{\mathbf{A}}_2^{†}{\mathbf{XB}}_2^{†}-{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}\right]\mathbf{F}=\mathbf{0}.$$ E7

In that case, the general solution of (1) can be expressed as the following,

$$\begin{array}{cc}\mathbf{X}=& {\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}+{\mathbf{L}}_{A_1}{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{L}}_T\left({\mathbf{A}}_2^{†}{\mathbf{C}}_2{\mathbf{B}}_2^{†}-{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}\right){\mathbf{B}}_2{\mathbf{B}}_2^{†}+{\mathbf{T}}^{†}\mathbf{T}\left({\mathbf{A}}_2^{†}{\mathbf{C}}_2{\mathbf{B}}_2^{†}-{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}\right){\mathbf{B}}_2{\mathbf{N}}^{†}{\mathbf{R}}_{B_1}\\ & +{\mathbf{L}}_{A_1}\left(\mathbf{Z}-{\mathbf{H}}^{†}{\mathbf{HZB}}_2{\mathbf{B}}_2^{†}\right)-{\mathbf{L}}_{A_1}{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{L}}_T{\mathbf{WNB}}_2^{†}+\left(\mathbf{W}-{\mathbf{T}}^{†}{\mathbf{\text{TWNN}}}^{†}\right)\times {\mathbf{R}}_{B_1},\end{array}$$ E8

where Z and W are the arbitrary matrices over $\mathbb{H}$ with compatible dimensions.

Some simplification of (8) can be derived due to the quaternionic analog of the following proposition.

Lemma 3.2. [32] If $\mathbf{A}\in {\mathbb{H}}^{n\times n}$ is Hermitian and idempotent, then the following equation holds for any matrix $\mathbf{B}\in {\mathbb{H}}^{m\times n}$ ,

$$\mathbf{A}{\left(\mathbf{BA}\right)}^{†}={\left(\mathbf{BA}\right)}^{†}.$$ E9

It is evident that if $\mathbf{A}\in {\mathbb{H}}^{n\times n}$ is Hermitian and idempotent, then the following equation is true as well,

$${\left(\mathbf{AB}\right)}^{†}\mathbf{A}={\left(\mathbf{AB}\right)}^{†}.$$ E10

Since ${\mathbf{L}}_{A_1}$ , ${\mathbf{R}}_{B_1}$ , and R_H are projectors, then using (9) and (10), we have, respectively,

$$\begin{array}{l}{\mathbf{L}}_{A_1}{\mathbf{H}}^{†}={\mathbf{L}}_{A_1}{\left({\mathbf{A}}_2{\mathbf{L}}_{A_1}\right)}^{†}={\left({\mathbf{A}}_2{\mathbf{L}}_{A_1}\right)}^{†}={\mathbf{H}}^{†},\\ {\mathbf{N}}^{†}{\mathbf{R}}_{B_1}={\left({\mathbf{R}}_{B_1}{\mathbf{B}}_2\right)}^{†}{\mathbf{R}}_{B_1}={\left({\mathbf{R}}_{B_1}{\mathbf{B}}_2\right)}^{†}={\mathbf{N}}^{†},\\ {\mathbf{T}}^{†}\mathbf{T}={\left({\mathbf{R}}_H{\mathbf{A}}_2\right)}^{†}{\mathbf{R}}_H{\mathbf{A}}_2={\left({\mathbf{R}}_H{\mathbf{A}}_2\right)}^{†}{\mathbf{A}}_2={\mathbf{T}}^{†}{\mathbf{A}}_2,\\ {\mathbf{L}}_T=\mathbf{I}-{\mathbf{T}}^{†}\mathbf{T}=\mathbf{I}-{\mathbf{T}}^{†}{\mathbf{A}}_2.\end{array}$$ E11

Using (11) and (6), we obtain the following expression of (8),

$$\begin{array}{l}\mathbf{X}={\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}+{\mathbf{H}}^{†}{\mathbf{A}}_2\left(\mathbf{I}-{\mathbf{T}}^{†}{\mathbf{A}}_2\right)\left({\mathbf{A}}_2^{†}{\mathbf{C}}_2{\mathbf{B}}_2^{†}-{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}\right){\mathbf{B}}_2{\mathbf{B}}_2^{†}\hfill \\ +{\mathbf{T}}^{†}{\mathbf{A}}_2\left({\mathbf{A}}_2^{†}{\mathbf{C}}_2{\mathbf{B}}_2^{†}-{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}\right){\mathbf{B}}_2{\mathbf{N}}^{†}+{\mathbf{L}}_{A_1}\left(\mathbf{Z}-{\mathbf{H}}^{†}{\mathbf{HZB}}_2{\mathbf{B}}_2^{†}\right)-{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{L}}_T{\mathbf{WNB}}_2^{†}\hfill \\ +\left(\mathbf{W}-{\mathbf{T}}^{†}{\mathbf{\text{TWNN}}}^{†}\right){\mathbf{R}}_{B_1}={\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}+{\mathbf{H}}^{†}{\mathbf{C}}_2{\mathbf{B}}_2^{†}+{\mathbf{H}}^{†}\left({\mathbf{A}}_2{\mathbf{T}}^{†}-\mathbf{I}\right){\mathbf{A}}_2{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}{\mathbf{Q}}_{B_2}\hfill \\ -{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{T}}^{†}{\mathbf{C}}_2{\mathbf{B}}_2^{†}+{\mathbf{T}}^{†}{\mathbf{C}}_2{\mathbf{N}}^{†}-{\mathbf{T}}^{†}{\mathbf{A}}_2{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}{\mathbf{B}}_2{\mathbf{N}}^{†}+{\mathbf{L}}_{A_1}\left(\mathbf{Z}-{\mathbf{H}}^{†}{\mathbf{HZB}}_2{\mathbf{B}}_2^{†}\right)\hfill \\ -{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{L}}_T{\mathbf{WNB}}_2^{†}+\left(\mathbf{W}-{\mathbf{T}}^{†}{\mathbf{\text{TWNN}}}^{†}\right){\mathbf{R}}_{B_1}.\hfill \end{array}$$ E12

By putting ${\mathbf{Z}}_1={\mathbf{W}}_1=\mathbf{0}$ in (12), the partial solution of (8) can be derived,

$$\begin{array}{ll}{\mathbf{X}}_0=& {\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}+{\mathbf{H}}^{†}{\mathbf{C}}_2{\mathbf{B}}_2^{†}+{\mathbf{T}}^{†}{\mathbf{C}}_2{\mathbf{N}}^{†}+{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{T}}^{†}{\mathbf{A}}_2{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}{\mathbf{Q}}_{B_2}\hfill \\ & -{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}{\mathbf{Q}}_{B_2}-{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{T}}^{†}{\mathbf{C}}_2{\mathbf{B}}_2^{†}-{\mathbf{T}}^{†}{\mathbf{A}}_2{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_1^{†}{\mathbf{B}}_2{\mathbf{N}}^{†}.\hfill \end{array}$$ E13

Further we give determinantal representations of (13). Let ${\mathbf{A}}_1=\left(a_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}_{r_1}^{m\times n}$ , ${\mathbf{B}}_1=\left(b_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}_{r_2}^{r\times s}$ , ${\mathbf{A}}_2=\left(a_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}_{r_3}^{k\times n}$ , ${\mathbf{B}}_2=\left(b_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}_{r_4}^{r\times p}$ , ${\mathbf{C}}_1=\left(c_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}^{m\times s}$ , and ${\mathbf{C}}_2=\left(c_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{k\times p}$ , and there exist ${\mathbf{A}}_1^{†}=\left(a_{\mathit{ij}}^{\left(1\right),†}\right)\in {\mathbb{H}}^{n\times m}$ , ${\mathbf{B}}_2^{†}=\left(b_{\mathit{ij}}^{\left(2\right),†}\right)\in {\mathbb{H}}^{p\times r}$ , ${\mathbf{H}}^{†}=\left(h_{\mathit{ij}}^{†}\right)\in {\mathbb{H}}^{n\times k}$ , ${\mathbf{N}}^{†}=\left(n_{\mathit{ij}}^{†}\right)\in {\mathbb{H}}^{p\times r}$ , and ${\mathbf{T}}^{†}=\left(t_{\mathit{ij}}^{†}\right)\in {\mathbb{H}}^{n\times k}$ . Let $\text{rank}\mathbf{H}=\text{min}\left\{\text{rank}{\mathbf{A}}_2,\text{rank}{\mathbf{L}}_{A_1}\right\}=r_5$ , $\text{rank}\mathbf{N}=\text{min}\left\{\text{rank}{\mathbf{B}}_2,\text{rank}{\mathbf{R}}_{B_1}\right\}=r_6$ , and $\text{rank}\mathbf{T}=\text{min}\left\{\text{rank}{\mathbf{A}}_2,\text{rank}{\mathbf{R}}_H\right\}=r_7$ . Consider each term of (13) separately.

1. (i) By Theorem 2.3 for the first term, $x_{\mathit{ij}}^{01}$ , of (13), we have

$$x_{\mathit{ij}}^{01}=\frac{\sum _{\beta \in J_{r_1,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right)}_{.i}\left({\mathbf{d}}_{.j}^{B_1}\right)\right)}_{\beta }^{\beta }}{\sum _{\beta \in J_{r_1,n}}{\left|{\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right|}_{\beta }^{\beta }\sum _{\alpha \in I_{r_2,p}}{\left|{\mathbf{B}}_1{\mathbf{B}}_1^{\ast }\right|}_{\alpha }^{\alpha }},$$ E14

or

$$x_{\mathit{ij}}^{01}=\frac{\sum _{\alpha \in I_{r_2,q}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_1{\mathbf{B}}_1^{\ast }\right)}_{j.}\left({\mathbf{d}}_{i.}^{A_1}\right)\right)}_{\alpha }^{\alpha }}{\sum _{\beta \in J_{r_1,p}}{\left|{\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right|}_{\beta }^{\beta }\sum _{\alpha \in I_{r_2,q}}{\left|{\mathbf{B}}_1{\mathbf{B}}_1^{\ast }\right|}_{\alpha }^{\alpha }},$$ E15

where

$${\mathbf{d}}_{.j}^{B_1}=\left[\sum _{\alpha \in I_{r_2,p}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_1{\mathbf{B}}_1^{\ast }\right)}_{j.}\left({\tilde{c}}_{q.}^{\left(1\right)}\right)\right)}_{\alpha }^{\alpha }\right]\in {\mathbb{H}}^{n\times 1},q=1,\dots ,n,$$

$${\mathbf{d}}_{i.}^{A_1}=\left[\sum _{\beta \in J_{r_1,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right)}_{.i}\left({\tilde{c}}_{.l}^{\left(1\right)}\right)\right)}_{\beta }^{\beta }\right]\in {\mathbb{H}}^{1\times r},l=1,\dots ,r,$$

are the column vector and the row vector, respectively. ${\tilde{\mathbf{c}}}_{q.}^{\left(1\right)}$ and ${\tilde{\mathbf{c}}}_{.l}^{\left(1\right)}$ are the qth row and the lth column of ${\tilde{\mathbf{C}}}_1={\mathbf{A}}_1^{\ast }{\mathbf{C}}_1{\mathbf{B}}_1^{\ast }$ .

1. (ii) Similarly, for the second term of (13), we have

$$x_{\mathit{ij}}^{02}=\frac{\sum _{\beta \in J_{r_5,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{H}}^{\ast }\mathbf{H}\right)}_{.i}\left({\mathbf{d}}_{.j}^{B_2}\right)\right)}_{\beta }^{\beta }}{\sum _{\beta \in J_{r_5,n}}{\left|{\mathbf{H}}^{\ast }\mathbf{H}\right|}_{\beta }^{\beta }\sum _{\alpha \in I_{r_4,r}}{\left|{\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right|}_{\alpha }^{\alpha }},$$ E16

or

$$x_{\mathit{ij}}^{02}=\frac{\sum _{\alpha \in I_{r_4,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right)}_{j.}\left({\mathbf{d}}_{i.}^H\right)\right)}_{\alpha }^{\alpha }}{\sum _{\beta \in J_{r_5,n}}{\left|{\mathbf{H}}^{\ast }\mathbf{H}\right|}_{\beta }^{\beta }\sum _{\alpha \in I_{r_4,r}}{\left|{\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right|}_{\alpha }^{\alpha }},$$ E17

where

$${\mathbf{d}}_{.j}^{B_2}=\left[\sum _{\alpha \in I_{r_4,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right)}_{j.}\left({\tilde{c}}_{q.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha }\right]\in {\mathbb{H}}^{n\times 1},q=1,\dots ,n,$$

$${\mathbf{d}}_{i.}^H=\left[\sum _{\beta \in J_{r_5,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{H}}^{\ast }\mathbf{H}\right)}_{.i}\left({\tilde{c}}_{.l}^{\left(2\right)}\right)\right)}_{\beta }^{\beta }\right]\in {\mathbb{H}}^{1\times r},l=1,\dots ,r,$$

are the column vector and the row vector, respectively. ${\tilde{\mathbf{c}}}_{q.}^{\left(2\right)}$ and ${\tilde{\mathbf{c}}}_{.l}^{\left(2\right)}$ are the qth row and the lth column of ${\tilde{\mathbf{C}}}_2={\mathbf{H}}^{\ast }{\mathbf{C}}_2{\mathbf{B}}_2^{\ast }$ . Note that ${\mathbf{H}}^{\ast }\mathbf{H}={\left({\mathbf{A}}_2{\mathbf{L}}_{A_1}\right)}^{\ast }{\mathbf{A}}_2{\mathbf{L}}_{A_1}={\mathbf{L}}_{A_1}{\mathbf{A}}_2^{\ast }{\mathbf{A}}_2{\mathbf{L}}_{A_1}.$

1. (iii) The third term of (13) can be obtained by Theorem 2.3 as well. Then

$$x_{\mathit{ij}}^{03}=\frac{\sum _{\beta \in J_{r_7,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{T}}^{\ast }\mathbf{T}\right)}_{.i}\left({\mathbf{d}}_{.j}^N\right)\right)}_{\beta }^{\beta }}{\sum _{\beta \in J_{r_7,n}}{\left|{\mathbf{T}}^{\ast }\mathbf{T}\right|}_{\beta }^{\beta }\sum _{\alpha \in I_{r_6,r}}{\left|{\mathbf{NN}}^{\ast }\right|}_{\alpha }^{\alpha }},$$ E18

or

$$x_{\mathit{ij}}^{03}=\frac{\sum _{\alpha \in I_{r_6,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{NN}}^{\ast }\right)}_{j.}\left({\mathbf{d}}_{i.}^T\right)\right)}_{\alpha }^{\alpha }}{\sum _{\beta \in J_{r_7,n}}{\left|{\mathbf{T}}^{\ast }\mathbf{T}\right|}_{\beta }^{\beta }\sum _{\alpha \in I_{r_6,r}}{\left|{\mathbf{NN}}^{\ast }\right|}_{\alpha }^{\alpha }},$$ E19

where

$${\mathbf{d}}_{.j}^N=\left[\sum _{\alpha \in I_{r_6,r}\left\{f\right\}}{\text{rdet}}_j{\left({\left({\mathbf{NN}}^{\ast }\right)}_{j.}\left({\widehat{c}}_{q.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha }\right]\in {\mathbb{H}}^{n\times 1},q=1,\dots ,n,$$

$${\mathbf{d}}_{i.}^T=\left[\sum _{\beta \in J_{r_7,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{T}}^{\ast }\mathbf{T}\right)}_{.i}\left({\widehat{c}}_{.l}^{\left(2\right)}\right)\right)}_{\beta }^{\beta }\right]\in {\mathbb{H}}^{1\times r},l=1,\dots ,r,$$

are the column vector and the row vector, respectively. ${\widehat{\mathbf{c}}}_{q.}^{\left(2\right)}$ is the qth row and ${\widehat{\mathbf{c}}}_{.l}^{\left(2\right)}$ is the lth column of ${\widehat{\mathbf{C}}}_2={\mathbf{T}}^{\ast }{\mathbf{C}}_2{\mathbf{N}}^{\ast }$ . The following expression gives some simplify in computing. Since ${\mathbf{T}}^{\ast }\mathbf{T}={\left({\mathbf{R}}_H{\mathbf{A}}_2\right)}^{\ast }={\mathbf{A}}_2^{\ast }{\mathbf{R}}_H^{\ast }{\mathbf{R}}_H{\mathbf{A}}_2={\mathbf{A}}_2^{\ast }{\mathbf{R}}_H{\mathbf{A}}_2$ and ${\mathbf{R}}_H=\mathbf{I}-{\mathbf{HH}}^{†}=\mathbf{I}-{\mathbf{A}}_2{\mathbf{L}}_{A_1}{\left({\mathbf{A}}_2{\mathbf{L}}_{A_1}\right)}^{†}=\mathbf{I}-{\mathbf{A}}_2{\left({\mathbf{A}}_2{\mathbf{L}}_{A_1}\right)}^{†}$ , then ${\mathbf{T}}^{\ast }\mathbf{T}={\mathbf{A}}_2^{\ast }\left(\mathbf{I}-{\mathbf{A}}_2{\left({\mathbf{A}}_2{\mathbf{L}}_{A_1}\right)}^{†}\right){\mathbf{A}}_2$ .

1. (iv) Using (3) for determinantal representations of H^† and T^† in the fourth term of (13), we obtain

$$x_{\mathit{ij}}^{04}=\frac{{\sum }_{q=1}^n{\sum }_{z=1}^n{\sum }_{f=1}^r\sum _{\beta \in J_{r_5,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{H}}^{\ast }\mathbf{H}\right)}_{.i}\left({\mathbf{a}}_{.q}^{\left(2,H\right)}\right)\right)}_{\beta }^{\beta }\sum _{\beta \in J_{r_7,n}\left\{q\right\}}{\text{cdet}}_q{\left({\left({\mathbf{T}}^{\ast }\mathbf{T}\right)}_{.q}\left({\mathbf{a}}_{.z}^{\left(2,T\right)}\right)\right)}_{\beta }^{\beta }{x}_{\mathit{zf}}^{01}{q}_{\mathit{fj}}}{\sum _{\beta \in J_{r_5,n}}{\left|{\mathbf{H}}^{\ast }\mathbf{H}\right|}_{\beta }^{\beta }\sum _{\beta \in J_{r_7,n}}{\left|{\mathbf{T}}^{\ast }\mathbf{T}\right|}_{\beta }^{\beta }},$$ E20

where ${\mathbf{a}}_{.i}^{\left(2,H\right)}$ and ${\mathbf{a}}_{.i}^{\left(2,T\right)}$ are the ith columns of the matrices H^*A₂ and T^*A₂, respectively; q_fj is the (fj)th element of ${\mathbf{Q}}_{B_2}$ with the determinantal representation,

$$q_{\mathit{fj}}=\frac{\sum _{\alpha \in I_{r_4,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right)}_{j.}\left({\ddot{\mathbf{b}}}_{f.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha }}{\sum _{\alpha \in I_{r_4,r}}{\left|{\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right|}_{\alpha }^{\alpha }},$$

and ${\ddot{\mathbf{b}}}_{f.}^{\left(2\right)}$ is the fth row of ${\mathbf{B}}_2{\mathbf{B}}_2^{\ast }$ . Note that ${\mathbf{H}}^{\ast }{\mathbf{A}}_2={\mathbf{L}}_{A_1}{\mathbf{A}}_2^{\ast }{\mathbf{A}}_2$ and ${\mathbf{T}}^{\ast }{\mathbf{A}}_2={\mathbf{A}}_2^{\ast }{\mathbf{R}}_H{\mathbf{A}}_2={\mathbf{A}}_2^{\ast }\left(\mathbf{I}-{\mathbf{A}}_2{\left({\mathbf{A}}_2{\mathbf{L}}_{A_1}\right)}^{†}\right){\mathbf{A}}_2$ .

1. (v) Similar to the previous case,

$$x_{\mathit{ij}}^{05}=\frac{{\sum }_{q=1}^n{\sum }_{f=1}^r\sum _{\beta \in J_{r_5,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{H}}^{\ast }\mathbf{H}\right)}_{.i}\left({\mathbf{a}}_{.q}^{\left(2,H\right)}\right)\right)}_{\beta }^{\beta }{x}_{\mathit{qf}}^{01}{q}_{\mathit{fj}}}{\sum _{\beta \in J_{r_5,n}}{\left|{\mathbf{H}}^{\ast }\mathbf{H}\right|}_{\beta }^{\beta }},$$ E21

1. (vi) Consider the sixth term by analogy to the fourth term. So,

$$x_{\mathit{ij}}^{06}=\frac{{\sum }_{q=1}^n\sum _{\beta \in J_{r_5,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{H}}^{\ast }\mathbf{H}\right)}_{.i}\left({\mathbf{a}}_{.q}^{\left(2,H\right)}\right)\right)}_{\beta }^{\beta }{\phi }_{\mathit{qj}}}{\sum _{\beta \in J_{r_5,n}}{\left|{\mathbf{H}}^{\ast }\mathbf{H}\right|}_{\beta }^{\beta }\sum _{\beta \in J_{r_7,n}}{\left|{\mathbf{T}}^{\ast }\mathbf{T}\right|}_{\beta }^{\beta }\sum _{\alpha \in I_{r_4,r}}{\left|{\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right|}_{\alpha }^{\alpha }},$$ E22

where

$${\phi }_{\mathit{qj}}=\sum _{\beta \in J_{r_7,n}\left\{i\right\}}{\text{cdet}}_q{\left({\left({\mathbf{T}}^{\ast }\mathbf{T}\right)}_{.q}\left({\psi }_{.j}^{B_2}\right)\right)}_{\beta }^{\beta },$$ E23

or

$${\phi }_{\mathit{qj}}=\sum _{\alpha \in I_{r_4,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right)}_{j.}\left({\psi }_{q.}^T\right)\right)}_{\alpha }^{\alpha },$$ E24

and

$${\psi }_{.j}^{B_2}=\left[\sum _{\alpha \in I_{r_4,r}\left\{f\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right)}_{j.}\left({\stackrel{⌣}{c}}_{q.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha }\right]\in {\mathbb{H}}^{1\times n},q=1,\dots ,n,$$

$${\psi }_{q.}^T=\left[\sum _{\beta \in J_{r_7,n}\left\{q\right\}}{\text{cdet}}_q{\left({\left({\mathbf{T}}^{\ast }\mathbf{T}\right)}_{.q}\left({\stackrel{⌣}{c}}_{.l}^{\left(2\right)}\right)\right)}_{\beta }^{\beta }\right]\in {\mathbb{H}}^{r\times 1},l=1,\dots ,r,$$

are the column vector and the row vector, respectively. ${\stackrel{⌣}{\mathbf{c}}}_{q.}^{\left(2\right)}$ and ${\stackrel{⌣}{\mathbf{c}}}_{.l}^{\left(2\right)}$ are the qth row and the lth column of ${\stackrel{⌣}{\mathbf{C}}}_2={\mathbf{T}}^{\ast }{\mathbf{C}}_2{\mathbf{B}}_2^{\ast }$ for all $i=1,\dots ,n$ and $j=1,\dots ,p$ .

1. (vii) Using (3) for determinantal representations of and T^† and (2) for N^† in the seventh term of (13), we obtain

$$x_{\mathit{ij}}^{07}=\frac{{\sum }_{q=1}^n{\sum }_{f=1}^r\sum _{\beta \in J_{r_7,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{T}}^{\ast }\mathbf{T}\right)}_{.i}\left({\mathbf{a}}_{.q}^{\left(2,T\right)}\right)\right)}_{\beta }^{\beta }{x}_{\mathit{qf}}^{01}\sum _{\alpha \in I_{r_6,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{NN}}^{\ast }\right)}_{j.}\left({\mathbf{b}}_{f.}^{\left(2,N\right)}\right)\right)}_{\alpha }^{\alpha }}{\sum _{\beta \in J_{r_7,n}}{\left|{\mathbf{T}}^{\ast }\mathbf{T}\right|}_{\beta }^{\beta }\sum _{\alpha \in I_{r_6,r}}{\left|{\mathbf{NN}}^{\ast }\right|}_{\alpha }^{\alpha }},$$ E25

where ${\mathbf{a}}_{.q}^{\left(2,T\right)}$ and ${\mathbf{b}}_{f.}^{\left(2,N\right)}$ are the qth column of T^*A₂ and the fth row of ${\mathbf{B}}_2{\mathbf{N}}^{\ast }={\mathbf{B}}_2{\mathbf{B}}_2^{\ast }{\mathbf{R}}_{B_1}$ , respectively.

Hence, we prove the following theorem.

Theorem 3.1. Let ${\mathbf{A}}_1\in {\mathbb{H}}_{r_1}^{m\times n}$ , ${\mathbf{B}}_1\in {\mathbb{H}}_{r_2}^{r\times s}$ , ${\mathbf{A}}_2\in {\mathbb{H}}_{r_3}^{k\times n}$ , ${\mathbf{B}}_2\in {\mathbb{H}}_{r_4}^{r\times p}$ , $\text{rank}\mathbf{H}=\text{rank}\left({\mathbf{A}}_2{\mathbf{L}}_{A_1}\right)=r_5$ , $\text{rank}\mathbf{N}=\left({\mathbf{R}}_{B_1}{\mathbf{B}}_2\right)=r_6$ , and $\text{rank}\mathbf{T}=\left({\mathbf{R}}_H{\mathbf{A}}_2\right)=r_7$ . Then, for the partial solution (13), ${\mathbf{X}}_0=\left(x_{\mathit{ij}}^0\right)\in {\mathbb{H}}^{n\times r}$ , of the system (1), we have,

$$x_{\mathit{ij}}^0=\sum _{\delta }{x}_{\mathit{ij}}^{0\delta },$$ E26

where the term $x_{\mathit{ij}}^{01}$ has the determinantal representations (14) and (15), $x_{\mathit{ij}}^{02}$ —(16) and (17), $x_{\mathit{ij}}^{03}$ —(18) and (19), $x_{\mathit{ij}}^{04}$ —(20), $x_{\mathit{ij}}^{05}$ —(21), $x_{\mathit{ij}}^{06}$ —(23) and (24), and $x_{\mathit{ij}}^{07}$ —(25).

4. 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 B₁ is vanished. Then, we have the system

$$\left\{\begin{array}{l}{\mathbf{A}}_1\mathbf{X}={\mathbf{C}}_1,\\ {\mathbf{A}}_2{\mathbf{XB}}_2={\mathbf{C}}_2.\end{array}\right.$$ E27

The following lemma is extended to matrices with quaternion entries.

Lemma 4.1. [7] Let ${\mathbf{A}}_1\in {\mathbb{H}}^{m\times n}$ , ${\mathbf{C}}_1\in {\mathbb{H}}^{m\times r}$ , ${\mathbf{A}}_2\in {\mathbb{H}}^{k\times n}$ , ${\mathbf{B}}_2\in {\mathbb{H}}^{r\times p}$ , and ${\mathbf{C}}_2\in {\mathbb{H}}^{k\times p}$ be given and $\mathbf{X}\in {\mathbb{H}}^{n\times r}$ is to be determined. Put $\mathbf{H}={\mathbf{A}}_2{\mathbf{L}}_{A_1}$ . Then, the following statements are equivalent:

1. System (27) is consistent.

2. ${\mathbf{R}}_{A_1}{\mathbf{C}}_1=\mathbf{0}$ , ${\mathbf{R}}_H\left({\mathbf{C}}_2-{\mathbf{A}}_2{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_2\right)=\mathbf{0}$ , ${\mathbf{C}}_2{\mathbf{L}}_{B_2}=\mathbf{0}$ .

3. $\text{rank}\left[{\mathbf{A}}_1{\mathbf{C}}_1\right]=\text{rank}\left[{\mathbf{A}}_1\right]$ , $\text{rank}\left[\begin{array}{l}{\mathbf{C}}_2\\ {\mathbf{B}}_2\end{array}\right]=\text{rank}\left[{\mathbf{B}}_2\right]$ , $\text{rank}\left[\begin{array}{ll}{\mathbf{A}}_1& {\mathbf{C}}_1{\mathbf{B}}_2\\ {\mathbf{A}}_2& {\mathbf{C}}_2\end{array}\right]=\text{rank}\left[\begin{array}{l}{\mathbf{A}}_1\\ {\mathbf{A}}_2\end{array}\right]$ .

In this case, the general solution of (27) can be expressed as

$$\mathbf{X}={\mathbf{A}}_1^{†}{\mathbf{C}}_1+{\mathbf{L}}_{A_1}{\mathbf{H}}^{†}\left({\mathbf{C}}_2-{\mathbf{A}}_2{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_2\right){\mathbf{B}}_2^{†}+{\mathbf{L}}_{A_1}{\mathbf{L}}_H{\mathbf{Z}}_1+{\mathbf{L}}_{A_1}{\mathbf{W}}_1{\mathbf{R}}_{B_2},$$ E28

where Z₁ and W₁ are the arbitrary matrices over $\mathbb{H}$ with appropriate sizes.

Since by (9), ${\mathbf{L}}_{A_1}{\mathbf{H}}^{†}={\mathbf{L}}_{A_1}{\left({\mathbf{A}}_2{\mathbf{L}}_{A_1}\right)}^{†}={\left({\mathbf{A}}_2{\mathbf{L}}_{A_1}\right)}^{†}={\mathbf{H}}^{†},$ then we have some simplification of (28),

$$\mathbf{X}={\mathbf{A}}_1^{†}{\mathbf{C}}_1+{\mathbf{H}}^{†}{\mathbf{C}}_2{\mathbf{B}}_2^{†}-{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_2{\mathbf{B}}_2^{†}+{\mathbf{L}}_{A_1}{\mathbf{L}}_H{\mathbf{Z}}_1+{\mathbf{L}}_{A_1}{\mathbf{W}}_1{\mathbf{R}}_{B_2}.$$

By putting Z₁=W₁=0, there is the following partial solution of (27),

$${\mathbf{X}}_0={\mathbf{A}}_1^{†}{\mathbf{C}}_1+{\mathbf{H}}^{†}{\mathbf{C}}_2{\mathbf{B}}_2^{†}-{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{B}}_2{\mathbf{B}}_2^{†}.$$ E29

Theorem 4.1. Let ${\mathbf{A}}_1=\left(a_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}_{r_1}^{m\times n}$ , ${\mathbf{A}}_2=\left(a_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}_{r_2}^{k\times n}$ , ${\mathbf{B}}_2=\left(b_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}_{r_3}^{r\times p}$ , ${\mathbf{C}}_1=\left(c_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}^{m\times r}$ , and ${\mathbf{C}}_2=\left(c_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{k\times p}$ , and there exist ${\mathbf{A}}_1^{†}=\left(a_{\mathit{ij}}^{\left(1\right),†}\right)\in {\mathbb{H}}^{n\times m}$ , ${\mathbf{B}}_2^{†}=\left(b_{\mathit{ij}}^{\left(2\right),†}\right)\in {\mathbb{H}}^{p\times r}$ , and ${\mathbf{H}}^{†}=\left(h_{\mathit{ij}}^{†}\right)\in {\mathbb{H}}^{n\times k}$ . Let $\text{rank}\mathbf{H}=\text{min}\left\{\text{rank}{\mathbf{A}}_2,\text{rank}{\mathbf{L}}_{A_1}\right\}=r_4$ . Denote ${\mathbf{A}}_1^{\ast }{\mathbf{C}}_1≕{\widehat{\mathbf{C}}}_1=\left({\widehat{c}}_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}^{n\times r}$ , ${\mathbf{H}}^{\ast }{\mathbf{C}}_2{\mathbf{B}}_2^{\ast }≕{\widehat{\mathbf{C}}}_2=\left({\widehat{c}}_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{n\times r}$ , ${\mathbf{H}}^{\ast }{\mathbf{A}}_2{\mathbf{A}}_1^{\ast }≕{\widehat{\mathbf{A}}}_2=\left({\widehat{a}}_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{n\times m}$ , and ${\mathbf{C}}_1{\mathbf{Q}}_{B_2}≕\widehat{\mathbf{Q}}=\left({\widehat{q}}_{\mathit{ij}}\right)\in {\mathbb{H}}^{m\times p}$ . Then, the partial solution (29), ${\mathbf{X}}_0=\left(x_{\mathit{ij}}^0\right)\in {\mathbb{H}}^{n\times r}$ , possesses the following determinantal representations,

$$\begin{array}{c}{x}_{\mathit{ij}}^0=\frac{{\sum }_{\beta \in J_{r_1,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right)}_{.i}\left({\widehat{\mathbf{c}}}_{.j}^{\left(1\right)}\right)\right)}_{\beta }^{\beta }}{{\sum }_{\beta \in J_{r_1,n}}{\left|{\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right|}_{\beta }^{\beta }}+\frac{d_{\mathit{ij}}^{\left(\lambda \right)}}{{\sum }_{\beta \in J_{r_4,n}}{\left|{\mathbf{H}}^{\ast }\mathbf{H}\right|}_{\beta }^{\beta }{\sum }_{\alpha \in I_{r_3,r}}{\left|{\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right|}_{\alpha }^{\alpha }}\\ -\frac{{\sum }_{l=1}^m{g}_{\mathit{il}}^{\left(\mu \right)}{\sum }_{\alpha \in I_{r_3,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right)}_{j.}\left({\widehat{\mathbf{q}}}_{l.}\right)\right)}_{\alpha }^{\alpha }}{{\sum }_{\beta \in J_{r_4,n}}{\left|{\mathbf{H}}^{\ast }\mathbf{H}\right|}_{\beta }^{\beta }{\sum }_{\alpha \in I_{r_1,m}}{\left|{\mathbf{A}}_1{\mathbf{A}}_1^{\ast }\right|}_{\alpha }^{\alpha }{\sum }_{\alpha \in I_{r_3,r}}{\left|{\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right|}_{\alpha }^{\alpha }}\end{array}$$

for all $\lambda =1,2$ and $\mu =1,2$ . Here

$$d_{\mathit{ij}}^{\left(1\right)}≔\sum _{\alpha \in I_{r_3,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right)}_{j.}\left({\mathbf{v}}_{i.}^{\left(1\right)}\right)\right)}_{\alpha }^{\alpha },g_{\mathit{il}}^{\left(1\right)}≔\sum _{\alpha \in I_{r_1,m}\left\{l\right\}}{\text{rdet}}_l{\left({\left({\mathbf{A}}_1{\mathbf{A}}_1^{\ast }\right)}_{l.}\left({\mathbf{u}}_{i.}^{\left(1\right)}\right)\right)}_{\alpha }^{\alpha },$$

and the row-vectors ${\mathbf{v}}_{i.}^{\left(1\right)}=\left(v_{i1}^{\left(1\right)},\dots ,v_{\mathit{ir}}^{\left(1\right)}\right)$ and ${\mathbf{u}}_{i.}^{\left(1\right)}=\left(u_{i1}^{\left(1\right)},\dots ,u_{\mathit{im}}^{\left(1\right)}\right)$ such that

$$v_{\mathit{it}}^{\left(1\right)}≔\sum _{\beta \in J_{r_4,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{H}}^{\ast }\mathbf{H}\right)}_{.i}\left({\widehat{\mathbf{c}}}_{.t}^{\left(2\right)}\right)\right)}_{\beta }^{\beta },u_{\mathit{iz}}^{\left(1\right)}≔\sum _{\beta \in J_{r_4,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{H}}^{\ast }\mathbf{H}\right)}_{.i}\left({\widehat{\mathbf{a}}}_{.z}^{\left(2\right)}\right)\right)}_{\beta }^{\beta }.$$

In another case,

$$d_{\mathit{ij}}^{\left(2\right)}≔\sum _{\beta \in J_{r_4,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{H}}^{\ast }\mathbf{H}\right)}_{.i}\left({\mathbf{v}}_{.j}^{\left(2\right)}\right)\right)}_{\beta }^{\beta },g_{\mathit{il}}^{\left(2\right)}≔\sum _{\beta \in J_{r_4,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{H}}^{\ast }\mathbf{H}\right)}_{.i}\left({\mathbf{u}}_{.l}^{\left(2\right)}\right)\right)}_{\beta }^{\beta }.$$

and the column-vectors ${\mathbf{v}}_{.j}^{\left(2\right)}=\left(v_{1j}^{\left(2\right)},\dots ,v_{\mathit{nj}}^{\left(2\right)}\right)$ and ${\mathbf{u}}_{.l}^{\left(2\right)}=\left(u_{1l}^{\left(2\right)},\dots ,u_{\mathit{nl}}^{\left(2\right)}\right)$ such that

$$v_{\mathit{qj}}^{\left(2\right)}≔\sum _{\alpha \in I_{r_3,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right)}_{j.}\left({\widehat{\mathbf{c}}}_{q.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha },u_{\mathit{ql}}^{\left(2\right)}≔\sum _{\alpha \in I_{r_1,m}\left\{l\right\}}{\text{rdet}}_l{\left({\left({\mathbf{A}}_1{\mathbf{A}}_1^{\ast }\right)}_{l.}\left(a_{q.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha }$$

Proof. The proof is similar to the proof of Theorem 3.1.

Let in Eq.(1), the matrix A₁ is vanished. Then, we have the system,

$$\left\{\begin{array}{l}{\mathbf{XB}}_1={\mathbf{C}}_1,\\ {\mathbf{A}}_2{\mathbf{XB}}_2={\mathbf{C}}_2.\end{array}\right.$$ E30

The following lemma is extended to matrices with quaternion entries as well.

Lemma 4.2. [7] Let ${\mathbf{B}}_1\in {\mathbb{H}}^{r\times s}$ , ${\mathbf{C}}_1\in {\mathbb{H}}^{n\times s}$ , ${\mathbf{A}}_2\in {\mathbb{H}}^{k\times n}$ , ${\mathbf{B}}_2\in {\mathbb{H}}^{r\times p}$ , and ${\mathbf{C}}_2\in {\mathbb{H}}^{k\times p}$ be given and $\mathbf{X}\in {\mathbb{H}}^{n\times r}$ is to be determined. Put $\mathbf{N}={\mathbf{R}}_{B_1}{\mathbf{B}}_2$ . Then, the following statements are equivalent:

1. System (30) is consistent.

2. ${\mathbf{R}}_{A_2}{\mathbf{C}}_2=\mathbf{0}$ , $\left({\mathbf{C}}_2-{\mathbf{A}}_2{\mathbf{C}}_1{\mathbf{B}}_1^{†}{\mathbf{B}}_2\right){\mathbf{L}}_N=\mathbf{0}$ , ${\mathbf{C}}_2{\mathbf{L}}_{B_2}=\mathbf{0}$ .

3. $\text{rank}\left[\begin{array}{ll}{\mathbf{A}}_2& {\mathbf{C}}_2\end{array}\right]=\text{rank}\left[{\mathbf{A}}_2\right]$ , $\text{rank}\left[\begin{array}{l}{\mathbf{C}}_1\\ {\mathbf{B}}_1\end{array}\right]=\text{rank}\left[{\mathbf{B}}_1\right]$ , $\text{rank}\left[\begin{array}{ll}{\mathbf{C}}_2& {\mathbf{A}}_2{\mathbf{C}}_1\\ {\mathbf{B}}_2& {\mathbf{B}}_1\end{array}\right]=\text{rank}\left[\begin{array}{ll}{\mathbf{B}}_2& {\mathbf{B}}_1\end{array}\right]$ .

In this case, the general solution of (30) can be expressed as

$$\mathbf{X}={\mathbf{C}}_1{\mathbf{B}}_1^{†}+{\mathbf{A}}_2^{†}\left({\mathbf{C}}_2-{\mathbf{A}}_2{\mathbf{C}}_1{\mathbf{B}}_1^{†}{\mathbf{B}}_2\right){\mathbf{N}}^{†}{\mathbf{R}}_{B_1}+{\mathbf{L}}_{A_2}{\mathbf{W}}_2{\mathbf{R}}_{B_1}+{\mathbf{Z}}_2{\mathbf{R}}_N{\mathbf{R}}_{B_1},$$ E31

where Z₂ and W₂ are the arbitrary matrices over $\mathbb{H}$ with appropriate sizes.

Since by (10), ${\mathbf{N}}^{†}{\mathbf{R}}_{B_1}={\left({\mathbf{R}}_{B_1}{\mathbf{B}}_2\right)}^{†}{\mathbf{R}}_{B_1}={\mathbf{N}}^{†},$ then some simplification of (31) can be derived,

$$\mathbf{X}={\mathbf{C}}_1{\mathbf{B}}_1^{†}+{\mathbf{A}}_2^{†}{\mathbf{C}}_2{\mathbf{N}}^{†}-{\mathbf{A}}_2{\mathbf{C}}_1{\mathbf{B}}_1^{†}{\mathbf{B}}_2{\mathbf{N}}^{†}+{\mathbf{L}}_{A_2}{\mathbf{W}}_2{\mathbf{R}}_{B_1}+{\mathbf{Z}}_2{\mathbf{R}}_N{\mathbf{R}}_{B_1}.$$

By putting Z₂=W₂=0, there is the following partial solution of (30),

$${\mathbf{X}}_0={\mathbf{C}}_1{\mathbf{B}}_1^{†}+{\mathbf{A}}_2^{†}{\mathbf{C}}_2{\mathbf{N}}^{†}-{\mathbf{A}}_2^{†}{\mathbf{A}}_2{\mathbf{C}}_1{\mathbf{B}}_1^{†}{\mathbf{B}}_2{\mathbf{N}}^{†}.$$ E32

The following theorem on determinantal representations of (29) can be proven similar to the proof of Theorem 3.1 as well.

Theorem 4.2. Let ${\mathbf{B}}_1=\left(b_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}_{r_1}^{r\times s}$ , ${\mathbf{A}}_2=\left(a_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}_{r_2}^{k\times n}$ , ${\mathbf{B}}_2=\left(b_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}_{r_3}^{r\times p}$ , ${\mathbf{C}}_1=\left(c_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}^{n\times s}$ , and ${\mathbf{C}}_2=\left(c_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{k\times p}$ , and there exist ${\mathbf{B}}_1^{†}=\left(b_{\mathit{ij}}^{\left(1\right),†}\right)\in {\mathbb{H}}^{s\times r}$ , ${\mathbf{A}}_2^{†}=\left(a_{\mathit{ij}}^{\left(2\right),†}\right)\in {\mathbb{H}}^{n\times k}$ , ${\mathbf{N}}^{†}=\left(n_{\mathit{ij}}^{†}\right)\in {\mathbb{H}}^{p\times r}$ . Let $\text{rank}\mathbf{N}=\text{min}\left\{\text{rank}{\mathbf{B}}_2,\text{rank}{\mathbf{R}}_{B_1}\right\}=r_4$ . Denote ${\mathbf{C}}_1{\mathbf{B}}_1^{\ast }≕{\tilde{\mathbf{C}}}_1=\left({\tilde{c}}_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}^{n\times r}$ , ${\mathbf{A}}_2^{\ast }{\mathbf{C}}_2{\mathbf{N}}^{\ast }≕{\tilde{\mathbf{C}}}_2=\left({\tilde{c}}_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{n\times r}$ , ${\mathbf{B}}_1^{\ast }{\mathbf{B}}_2{\mathbf{N}}^{\ast }≕{\tilde{\mathbf{B}}}_2=\left({\tilde{b}}_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{s\times r}$ , and ${\mathbf{P}}_{A_2}{\mathbf{C}}_1≕\tilde{\mathbf{P}}=\left({\tilde{p}}_{\mathit{ij}}\right)\in {\mathbb{H}}^{n\times s}$ . Then, the partial solution (32), ${\mathbf{X}}_0=\left(x_{\mathit{ij}}^0\right)\in {\mathbb{H}}^{n\times r}$ , possesses the following determinantal representations,

$$\begin{array}{c}{x}_{\mathit{ij}}^0=\frac{{\sum }_{\alpha \in I_{r_1,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_1{\mathbf{B}}_1^{\ast }\right)}_{j.}\left({\tilde{\mathbf{c}}}_{i.}^{\left(1\right)}\right)\right)}_{\alpha }^{\alpha }}{{\sum }_{\alpha \in I_{r_1,r}}{\left|{\mathbf{B}}_1{\mathbf{B}}_1^{\ast }\right|}_{\alpha }^{\alpha }}+\frac{d_{\mathit{ij}}^{\left(\lambda \right)}}{{\sum }_{\beta \in J_{r_2,n}}{\left|{\mathbf{A}}_2^{\ast }{\mathbf{A}}_2\right|}_{\beta }^{\beta }{\sum }_{\alpha \in I_{r_4,r}}{\left|{\mathbf{NN}}^{\ast }\right|}_{\alpha }^{\alpha }}\\ -\frac{{\sum }_{z=1}^s{\sum }_{\beta \in J_{r_2,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}_2^{\ast }{\mathbf{A}}_2\right)}_{.i}\left({\tilde{\mathbf{p}}}_{.z}\right)\right)}_{\beta }^{\beta }{g}_{\mathit{zj}}^{\left(\mu \right)}}{{\sum }_{\beta \in J_{r_2,n}}{\left|{\mathbf{A}}_2^{\ast }{\mathbf{A}}_2\right|}_{\beta }^{\beta }{\sum }_{\beta \in J_{r_1,s}}{\left|{\mathbf{B}}_1^{\ast }{\mathbf{B}}_1\right|}_{\beta }^{\beta }{\sum }_{\alpha \in I_{r_4,r}}{\left|{\mathbf{NN}}^{\ast }\right|}_{\alpha }^{\alpha }}\end{array}$$

for all $\lambda =1,2$ and $\mu =1,2$ . Here

$$d_{\mathit{ij}}^{\left(1\right)}≔\sum _{\alpha \in I_{r_3,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{NN}}^{\ast }\right)}_{j.}\left({\phi }_{i.}^{\left(1\right)}\right)\right)}_{\alpha }^{\alpha },g_{\mathit{il}}^{\left(1\right)}≔\sum _{\alpha \in I_{r_4,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{NN}}^{\ast }\right)}_{j.}\left({\psi }_{z.}\right)\right)}_{\alpha }^{\alpha },$$

and the row-vectors ${\phi }_{i.}^{\left(1\right)}=\left({\phi }_{i1}^{\left(1\right)},\dots ,{\phi }_{\mathit{ir}}^{\left(1\right)}\right)$ and ${\psi }_{i.}^{\left(1\right)}=\left({\psi }_{z1}^{\left(1\right)},\dots ,{\psi }_{\mathit{zr}}^{\left(1\right)}\right)$ such that

$${\phi }_{\mathit{it}}^{\left(1\right)}=\sum _{\beta \in J_{r_2,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}_2^{\ast }{\mathbf{A}}_2\right)}_{.i}\left(c_{.t}^{\left(2\right)}\right)\right)}_{\beta }^{\beta },{\psi }_{\mathit{zv}}^{\left(1\right)}=\sum _{\beta \in J_{r_1,n}\left\{z\right\}}{\text{cdet}}_z{\left({\left({\mathbf{B}}_1^{\ast }{\mathbf{B}}_1\right)}_{.i}\left(b_{.v}^{\left(2\right)}\right)\right)}_{\beta }^{\beta }.$$

In another case,

$$d_{\mathit{ij}}^{\left(2\right)}≔\sum _{\beta \in J_{r_2,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}_2^{\ast }{\mathbf{A}}_2\right)}_{.i}\left({\phi }_{.j}^{\left(2\right)}\right)\right)}_{\beta }^{\beta },g_{\mathit{zj}}^{\left(2\right)}≔\sum _{\beta \in J_{r_1,n}\left\{z\right\}}{\text{cdet}}_z{\left({\left({\mathbf{B}}_1^{\ast }{\mathbf{B}}_1\right)}_{.z}\left({\psi }_{.j}^{\left(2\right)}\right)\right)}_{\beta }^{\beta },$$

and the column-vectors ${\phi }_{.j}^{\left(2\right)}=\left({\phi }_{1j}^{\left(2\right)},\dots ,{\phi }_{\mathit{nj}}^{\left(2\right)}\right)$ and ${\psi }_{.j}^{\left(2\right)}=\left({\psi }_{1j}^{\left(2\right)},\dots ,{\psi }_{\mathit{sj}}^{\left(2\right)}\right)$ such that

$${\phi }_{\mathit{qj}}^{\left(2\right)}=\sum _{\alpha \in I_{r_4,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{NN}}^{\ast }\right)}_{j.}\left(c_{q.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha },{\psi }_{\mathit{uj}}^{\left(2\right)}≔\sum _{\alpha \in I_{r_4,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{NN}}^{\ast }\right)}_{j.}\left(b_{u.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha }.$$

Now, suppose that the both equations of (1) are one-sided. Let in Eq.(1), the matrices B₁ and A₂ are vanished. Then, we have the system

$$\left\{\begin{array}{l}{\mathbf{A}}_1\mathbf{X}={\mathbf{C}}_1,\\ {\mathbf{XB}}_2={\mathbf{C}}_2.\end{array}\right.$$ E33

The following lemma is extended to matrices with quaternion entries.

Lemma 4.3. [31] Let ${\mathbf{A}}_1\in {\mathbb{H}}^{m\times n}$ , ${\mathbf{B}}_2\in {\mathbb{H}}^{r\times p}$ , ${\mathbf{C}}_1\in {\mathbb{H}}^{m\times r}$ , and ${\mathbf{C}}_2\in {\mathbb{H}}^{n\times p}$ be given and $\mathbf{X}\in {\mathbb{H}}^{n\times r}$ is to be determined. Then, the system (33) is consistent if and only if ${\mathbf{R}}_{A_1}{\mathbf{C}}_1=\mathbf{0}$ , ${\mathbf{C}}_2{\mathbf{L}}_{B_2}=\mathbf{0}$ , and A₁C₂=C₁B₂. Under these conditions, the general solution to (33) can be established as

$$\mathbf{X}={\mathbf{A}}_1^{†}{\mathbf{C}}_1+{\mathbf{L}}_{A_1}{\mathbf{C}}_2{\mathbf{B}}_2^{†}+{\mathbf{L}}_{A_1}{\mathbf{UR}}_{B_2},$$ E34

where U is a free matrix over $\mathbb{H}$ with a suitable shape.

Due to the consistence conditions, Eq. (34) can be expressed as follows:

$$\begin{array}{c}\mathbf{X}={\mathbf{C}}_2{\mathbf{B}}_2^{†}+{\mathbf{A}}_1^{†}\left({\mathbf{C}}_1-{\mathbf{A}}_1{\mathbf{C}}_2{\mathbf{B}}_2^{†}\right)+{\mathbf{L}}_{A_1}{\mathbf{UR}}_{B_2}\\ ={\mathbf{C}}_2{\mathbf{B}}_2^{†}+{\mathbf{A}}_1^{†}\left({\mathbf{C}}_1-{\mathbf{C}}_1{\mathbf{B}}_2{\mathbf{B}}_2^{†}\right)+{\mathbf{L}}_{A_1}{\mathbf{UR}}_{B_2}={\mathbf{C}}_2{\mathbf{B}}_2^{†}+{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{R}}_{B_2}+{\mathbf{L}}_{A_1}{\mathbf{UR}}_{B_2},\end{array}$$

Consequently, the partial solution X⁰ to (33) is given by

$${\mathbf{X}}^0={\mathbf{A}}_1^{†}{\mathbf{C}}_1+{\mathbf{L}}_{A_1}{\mathbf{C}}_2{\mathbf{B}}_2^{†},$$ E35

or

$${\mathbf{X}}^0={\mathbf{C}}_2{\mathbf{B}}_2^{†}+{\mathbf{A}}_1^{†}{\mathbf{C}}_1{\mathbf{R}}_{B_2}.$$ E36

Due to the expression (35), the following theorem can be proven similar to the proof of Theorem 3.1.

Theorem 4.3. Let ${\mathbf{A}}_1=\left(a_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}_{r_1}^{m\times n}$ , ${\mathbf{B}}_2=\left(b_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}_{r_2}^{r\times p}$ , ${\mathbf{C}}_1=\left(c_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}^{m\times r}$ , and ${\mathbf{C}}_2=\left(c_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{n\times r}$ , and there exist ${\mathbf{A}}_1^{†}=\left(a_{\mathit{ij}}^{\left(1\right),†}\right)\in {\mathbb{H}}^{n\times m}$ , ${\mathbf{B}}_2^{†}=\left(b_{\mathit{ij}}^{\left(2\right),†}\right)\in {\mathbb{H}}^{p\times r}$ , and ${\mathbf{L}}_{A_1}=\mathbf{I}-{\mathbf{A}}_1^{†}{\mathbf{A}}_1≕\left(l_{\mathit{ij}}\right)\in {\mathbb{H}}^{n\times n}$ . Denote ${\mathbf{A}}_1^{\ast }{\mathbf{C}}_1≕{\widehat{\mathbf{C}}}_1=\left({\widehat{c}}_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}^{n\times r}$ and ${\mathbf{L}}_{A_1}{\mathbf{C}}_2{\mathbf{B}}_2^{\ast }≕{\widehat{\mathbf{C}}}_2=\left({\widehat{c}}_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{n\times r}$ . Then, the partial solution (35), ${\mathbf{X}}^0=\left(x_{\mathit{ij}}^0\right)\in {\mathbb{H}}^{n\times s}$ , possesses the following determinantal representation,

$$x_{\mathit{ij}}^0=\frac{\sum _{\beta \in J_{r_1,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right)}_{.i}\left({\widehat{\mathbf{c}}}_{.j}^{\left(1\right)}\right)\right)}_{\beta }^{\beta }}{\sum _{\beta \in J_{r_1,n}}{\left|{\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right|}_{\beta }^{\beta }}+\frac{\sum _{\alpha \in I_{r_2,r}\left\{j\right\}}{\text{rdet}}_j{\left({\left({\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right)}_{j.}\left({\widehat{\mathbf{c}}}_{i.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha }}{\sum _{\alpha \in I_{r_2,r}}{\left|{\mathbf{B}}_2{\mathbf{B}}_2^{\ast }\right|}_{\alpha }^{\alpha }},$$ E37

where ${\widehat{\mathbf{c}}}_{.j}^{\left(1\right)}$ is the jth column of ${\widehat{\mathbf{C}}}_1$ and ${\widehat{\mathbf{c}}}_{i.}^{\left(2\right)}$ is the ith row of ${\widehat{\mathbf{C}}}_2$ .

Remark 4.1. In accordance to the expression (36), we obtain the same representations, but with the denotations, ${\mathbf{C}}_2{\mathbf{B}}_2^{\ast }≕{\widehat{\mathbf{C}}}_2=\left({\widehat{c}}_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{n\times r}$ and ${\mathbf{A}}_1^{\ast }{\mathbf{C}}_1{\mathbf{R}}_{B_2}≕{\widehat{\mathbf{C}}}_1=\left({\widehat{c}}_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{n\times r}$ .

Let in Eq.(1), the matrices B₁ and B₂ are vanished. Then, we have the system

$$\left\{\begin{array}{l}{\mathbf{A}}_1\mathbf{X}={\mathbf{C}}_1,\\ {\mathbf{A}}_2\mathbf{X}={\mathbf{C}}_2.\end{array}\right.$$ E38

Lemma 4.4. [7] Suppose that ${\mathbf{A}}_1\in {\mathbb{H}}^{m\times n}$ , ${\mathbf{C}}_1\in {\mathbb{H}}^{m\times r}$ , ${\mathbf{A}}_2\in {\mathbb{H}}^{k\times n}$ , and ${\mathbf{C}}_2\in {\mathbb{H}}^{k\times r}$ are known and $\mathbf{X}\in {\mathbb{H}}^{n\times r}$ is unknown, $\mathbf{H}={\mathbf{A}}_2{\mathbf{L}}_{A_1}$ , $\mathbf{T}={\mathbf{R}}_H{\mathbf{A}}_2$ . Then, the system (38) is consistent if and only if ${\mathbf{A}}_i{\mathbf{A}}_i^{†}{\mathbf{C}}_i={\mathbf{C}}_i,$ for all $i=1,2$ and $\mathbf{T}\left({\mathbf{A}}_2^{†}{\mathbf{C}}_2-{\mathbf{A}}_1^{†}{\mathbf{C}}_1\right)=\mathbf{0}$ . Under these conditions, the general solution to (38) can be established as

$$\mathbf{X}={\mathbf{A}}_1^{†}{\mathbf{C}}_1+{\mathbf{L}}_{A_1}{\mathbf{H}}^{†}{\mathbf{A}}_2\left({\mathbf{A}}_2^{†}{\mathbf{C}}_2-{\mathbf{A}}_1^{†}{\mathbf{C}}_1\right)+{\mathbf{L}}_{A_1}{\mathbf{L}}_H\mathbf{Y},$$ E39

where Y is an arbitrary matrix over $\mathbb{H}$ with an appropriate size.

Using (10) and the consistency conditions, we simplify (39) accordingly, ${\mathbf{X}}^0={\mathbf{A}}_1^{†}{\mathbf{C}}_1+{\mathbf{H}}^{†}{\mathbf{C}}_2-{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{A}}_1^{†}{\mathbf{C}}_1+{\mathbf{L}}_{A_1}{\mathbf{L}}_H\mathbf{Y}.$ Consequently, the following partial solution of (39) will be considered

$${\mathbf{X}}^0={\mathbf{A}}_1^{†}{\mathbf{C}}_1+{\mathbf{H}}^{†}{\mathbf{C}}_2-{\mathbf{H}}^{†}{\mathbf{A}}_2{\mathbf{A}}_1^{†}{\mathbf{C}}_1.$$ E40

In the following theorem, we give the determinantal representations of (40).

Theorem 4.4. Let ${\mathbf{A}}_1=\left(a_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}_{r_1}^{m\times n}$ , ${\mathbf{A}}_2=\left(a_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}_{r_2}^{k\times n}$ , ${\mathbf{C}}_1=\left(c_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}^{m\times r}$ , ${\mathbf{C}}_2=\left(c_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{k\times r}$ , and there exist ${\mathbf{A}}_1^{†}=\left(a_{\mathit{ij}}^{\left(1\right),†}\right)\in {\mathbb{H}}^{n\times m}$ , ${\mathbf{H}}_2^{†}=\left(h_{\mathit{ij}}^{†}\right)\in {\mathbb{H}}^{n\times s}$ . Let $\text{rank}\mathbf{H}=\text{min}\left\{\text{rank}{\mathbf{A}}_2,\text{rank}{\mathbf{L}}_{A_1}\right\}=r_3$ . Denote ${\mathbf{A}}_1^{\ast }{\mathbf{C}}_1≕{\widehat{\mathbf{C}}}_1=\left({\widehat{c}}_{\mathit{ij}}^{\left(1\right)}\right)\in {\mathbb{H}}^{n\times r}$ , ${\mathbf{H}}^{\ast }{\mathbf{C}}_2≕{\widehat{\mathbf{C}}}_2=\left({\widehat{c}}_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{n\times r}$ , and ${\mathbf{H}}^{\ast }{\mathbf{A}}_2≕{\widehat{\mathbf{A}}}_2=\left({\widehat{a}}_{\mathit{ij}}^{\left(2\right)}\right)\in {\mathbb{H}}^{n\times n}$ . Then, ${\mathbf{X}}^0=\left(x_{\mathit{ij}}^0\right)\in {\mathbb{H}}^{n\times r}$ possesses the following determinantal representation,

$$\begin{array}{c}{x}_{\mathit{ij}}^0=\frac{{\sum }_{\beta \in J_{r_1,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right)}_{.i}\left({\widehat{\mathbf{c}}}_{.j}^{\left(1\right)}\right)\right)}_{\beta }^{\beta }}{{\sum }_{\beta \in J_{r_1,n}}{\left|{\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right|}_{\beta }^{\beta }}+\frac{{\sum }_{\beta \in J_{r_3,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{H}}^{\ast }\mathbf{H}\right)}_{.i}\left({\widehat{\mathbf{c}}}_{.j}^{\left(2\right)}\right)\right)}_{\beta }^{\beta }}{{\sum }_{\beta \in J_{r_3,n}}{\left|{\mathbf{H}}^{\ast }\mathbf{H}\right|}_{\beta }^{\beta }}\\ -\sum _{l=1}^n\frac{{\sum }_{\beta \in J_{r_3,n}\left\{i\right\}}{\text{cdet}}_i{\left({\left({\mathbf{H}}^{\ast }\mathbf{H}\right)}_{.i}\left({\widehat{\mathbf{a}}}_{.l}^{\left(2\right)}\right)\right)}_{\beta }^{\beta }}{{\sum }_{\beta \in J_{r_3,n}}{\left|{\mathbf{H}}^{\ast }\mathbf{H}\right|}_{\beta }^{\beta }}\cdot \frac{{\sum }_{\beta \in J_{r_1,n}\left\{l\right\}}{\text{cdet}}_l{\left({\left({\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right)}_{.l}\left({\widehat{\mathbf{c}}}_{.j}^{\left(1\right)}\right)\right)}_{\beta }^{\beta }}{{\sum }_{\beta \in J_{r_1,n}}{\left|{\mathbf{A}}_1^{\ast }{\mathbf{A}}_1\right|}_{\beta }^{\beta }},\end{array}$$ E41

where ${\widehat{\mathbf{c}}}_{.j}^{\left(1\right)}$ , ${\widehat{\mathbf{c}}}_{.j}^{\left(2\right)}$ , and ${\widehat{\mathbf{a}}}_{.j}^{\left(2\right)}$ are the jth columns of the matrices ${\widehat{\mathbf{C}}}_1$ , ${\widehat{\mathbf{C}}}_2$ , and ${\widehat{\mathbf{A}}}_2$ , respectively.

Proof. The proof is similar to the proof of Theorem 3.1.

5. 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 A₁XB₁=C₁, A₂XB₂=C₂, 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.

Conflict of interest

The author declares that there are no conflict interests.