Chapter 1 - Linear Equations and Matrices

Abstract

This chapter provides an overview of linear equations and matrices. It discusses linear systems that can be solved by the method of elimination and presents some examples. The method of elimination is adequate for the solution of linear systems when m and n are small. However, more sophisticated methods are required if m and n are large and for theoretical investigations. The chapter explains theory of matrices and defines the product of matrices. Every linear system has an augmented matrix, and, conversely, any matrix with more than one column is the augmented matrix of a linear system. The advantage of the augmented matrix is its brevity. The chapter considers matrix operations that provide efficient methods for the solution of linear systems and are also of interest in other applications. It discusses matrix multiplications, matrices in row echelon and reduced row echelon form, the inverse of a matrix, and elementary matrices, which can be used to find the inverse of an arbitrary nonsingular matrix. The chapter describes the properties of determinants of arbitrary order and discusses permutations with examples.