# Introduction to Matrix Algebra

*2022-08-16*

# INTRODUCTION TO MATRIX ALGEBRA

AUTAR KAW

*There is nothing noble about being superior*

*to another man; the true nobility lies in being*

*superior to your previous self* - Upanishads.

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Autar Kaw

University of South Florida

To Sherrie, Candace and Angelie

#### 0.0.0.2 PREFACE

This book is an extended primer for undergraduate Matrix Algebra. The book is either to be used as a refresher material for students who have already taken a course in Matrix Algebra or as a just-in-time tool if the burden of teaching Matrix Algebra has been distributed to several courses. In my own department, the Linear Algebra course was dropped from the curriculum a decade ago. It is now taught just in time in courses like Statics, Programming Concepts, Vibrations, and Controls.

The book is divided into ten chapters. Chapter 1 defines the matrix and introduces you to special types of matrices. Chapter 2 deals with vectors and shows how to add and subtract vectors, find linear combination of vectors and how to find the rank of a set of vectors. Two types of operations are done on matrices. Chapter 3 explains binary operations of matrices such as adding, subtracting, multiplying and inverse of matrices. Chapter 4 describes the unary matrix operations such as transpose, trace, and determinant of matrices. Chapter 5 combines the concepts of the previous chapters to show how simultaneous linear equations are set up in a matrix form. The concept of the inverse of a matrix is introduced. The classification of a system of equations into a consistent (unique or multiple solutions) and inconsistent solutions (no solution) is established.

Chapters 6-8 deal with the numerical methods of solving simultaneous linear equations. Chapter 6 shows a direct method of solving equations such as the Gaussian elimination methods, Chapter 7 illustrates the iterative method of Gauss-Seidel method and Chapter 8 explains the LU Decomposition method.

Chapter 9 shows how a reader can determine that a solution to the simultaneous linear equations is adequate through the concept of condition number.

The last chapter - Chapter 10 discusses the concepts of eigenvalues and eigenvectors of a square matrix.

All chapters open with the objectives (what you will learn) followed by the content and numerous examples. At the end of each chapter, hyperlinked key terms, exercise set and answers to selected problems are given.

The book will be **available soon** as a printed book in perfect
binding. For details about the price, please visit
http://www.autarkaw.com/books/matrixalgebra/index.html.

If you have any comments for the author, please send them to autarkaw@yahoo.com.

I would like to thank Eric Marvella, Sri Harsha Garapati and Lauren Kintner for their help in formatting the book. I would like to thank Matt Emmons and Luke Snyder for developing and proofreading the solutions manual for the book, respectively. I want to thank Bharat Pulaparthi and Jonas Fernandes on converting the textbook into the markdown format and putting it in this final format. The suggestions given by many users of the book have improved the quality and accuracy of the book.

I would like to thank my spouse, Sherrie and our children Candace and Angelie, who have encouraged me throughout the project.

Book website:
http://autarkaw.com/books/matrixalgebra/index.html

Open Education Resources for Matrix Algebra:
https://ma.MathForCollege.com