**Applications of Matrices**

different structures and also some questions and conjectures are posed. Finally, qualitative remarks about differential equation system solutions and their stability or asymptotical sta‐ bility are included. In Chapter 4, special compound ´ magic squares are considered and a dimensional subspace of the nullspace of the ´ squares is determined. All vectors in the subspaces possess the property that the sum of all entries of each vector equals zero. In Chapter 5, a new type of regular matrix generated by Fibonacci numbers is introduced and we shall investigate its various topological properties. The concept of mathematical regulari‐

**Prof. Dr. Hassan A. Yasser**

Thi-Qar University Science College Physics Department

Iraq

ty in terms of Fibonacci numbers and phyllotaxy have been discussed.

VIII Preface

**Chapter 1**

(1)

Provisional chapter

**Cramer's Rules for the System of Two-Sided Matrix**

DOI: 10.5772/intechopen.74105

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=C<sup>1</sup> 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=C<sup>2</sup> 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.

Keywords: Moore-Penrose inverse, quaternion matrix, Cramer rule, system matrix

The study of matrix equations and systems of matrix equations is an active research topic in

A1XB<sup>1</sup> ¼ C1, A2XB<sup>2</sup> ¼ C2:

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.

matrix theory and its applications. The system of classical two-sided matrix equations

Cramer's Rules for the System of Two-Sided Matrix

**Equations and of Its Special Cases**

Equations and of Its Special Cases

Additional information is available at the end of the chapter

2000 AMS subject classifications: 15A15, 16 W10

Additional information is available at the end of the chapter

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

Ivan I. Kyrchei

Ivan I. Kyrchei

Abstract

equations

1. Introduction

## **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

DOI: 10.5772/intechopen.74105

Ivan I. Kyrchei Ivan I. Kyrchei

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

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