Elementary Linear Algebra, 11th Edition

Chapter 2Determinants

CHAPTER CONTENTS

1. 2.1 Determinants by Cofactor Expansion
2. 2.2 Evaluating Determinants by Row Reduction
3. 2.3 Properties of Determinants; Cramer's Rule

INTRODUCTION

In this chapter we will study “determinants” or, more precisely, “determinant functions.” Unlike real-valued functions, such as $f(x)=x^2$, that assign a real number to a real variable $x$, determinant functions assign a real number $f(A)$ to a matrix variable $A$. Although determinants first arose in the context of solving systems of linear equations, they are rarely used for that purpose in real-world applications. While they can be useful for solving very small linear systems (say two or three unknowns), our main interest in them stems from the fact that they link together various concepts in linear algebra and provide a useful formula for the inverse of a matrix.

2.1 Determinants by Cofactor Expansion

In this section we will define the notion of a “determinant.” This will enable us to develop a specific formula for the inverse of an invertible matrix, whereas up to now we have had only a computational procedure for finding it. This, in turn, will eventually provide us with a formula for solutions of certain kinds of linear systems.

Recall from Theorem 1.4.5 that the $2\times 2$ matrix

$$A=\left[\begin{array}{ll}a\hfill & b\hfill \\ c\hfill & d\hfill \end{array}\right]$$

is invertible if and only if $ad-bc\ne 0$ and that the expression $ad-bc$ is called the determinant ...