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$$x$$f(A)$$A$

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 $2\times 2$

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

is invertible if and only if $ad-bc\ne 0$$ad-bc$determinant of the matrix $A$

(1)$$\det (A)=ad-bc\text{ or }\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$$

and that the inverse of $A$

(2)$$A^{-1}=\frac{1}{ \det (A)}\left[\begin{array}{ll}d\hfill & -b\hfill \\ -c\hfill & a\hfill \end{array}\right]$$

WARNING It is important to keep in mind that $\det (A)$number, whereas $A$matrix.

Minors and Cofactors

One of our main goals in this chapter is to obtain an analog of Formula (2) that is applicable to square matrices of all orders. For this purpose we will find it convenient to use subscripted entries when writing matrices or determinants. Thus, if we denote a $2\times 2$

$$A=[\begin{array}{l}{a}_{}\hfill \end{array}$$