Elementary Linear Algebra, 11th Edition

Chapter 1Systems of Linear Equations and Matrices

CHAPTER CONTENTS

• 1.1 Introduction to Systems of Linear Equations
• 1.2 Gaussian Elimination
• 1.3 Matrices and Matrix Operations
• 1.4 Inverses; Algebraic Properties of Matrices
• 1.5 Elementary Matrices and a Method for Finding $A^{-1}$
• 1.6 More on Linear Systems and Invertible Matrices
• 1.7 Diagonal,Triangular, and Symmetric Matrices
• 1.8 MatrixTransformations
• 1.9 Applications of Linear Systems
  • Network Analysis (Traffic Flow)
  • Electrical Circuits
  • Balancing Chemical Equations
  • Polynomial Interpolation
• 1.10 Leontief Input-Output Models

INTRODUCTION

Information in science, business, and mathematics is often organized into rows and columns to form rectangular arrays called “matrices” (plural of “matrix”). Matrices often appear as tables of numerical data that arise from physical observations, but they occur in various mathematical contexts as well. For example, we will see in this chapter that all of the information required to solve a system of equations such as

$$\begin{array}{rrr}\hfill 5x+y& \hfill =& \hfill 3\\ \hfill 2x-y& \hfill =& \hfill 4\end{array}$$

is embodied in the matrix

$$\left[\begin{array}{rrr}\hfill 5& \hfill 1& \hfill 3\\ \hfill 2& \hfill -1& \hfill 4\end{array}\right]$$

and that the solution of the system can be obtained by performing appropriate operations on this matrix. This is particularly important in developing computer programs for solving systems of equations because computers are well suited for manipulating arrays of numerical information. However, matrices are not simply a notational ...