Mathematics · Ch 4 — Determinants

Introduction

4.1

Introduction

4.1 Introduction

The study of determinants begins with a question: when does a system of linear equations have a unique solution? For a pair of equations in two variables,

\begin{aligned} a_1 x + b_1 y &= c_1 \\ a_2 x + b_2 y &= c_2 \end{aligned}

the matrix form is

\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}.

Here $\mathbf{A} = \begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix}$ is the coefficient matrix, and the system has a unique solution precisely when $a_1 b_2 - a_2 b_1 \neq 0$ . This number is so fundamental that it is called the determinant of $\mathbf{A}$ , written $\det \mathbf{A}$ .

Note

The name is apt: this number determines whether a unique solution exists. If it is zero, the system has either no solution or infinitely many solutions.

Determinants appear across engineering, science, economics, and social science. In this chapter we study determinants of matrices up to order $3 \times 3$ with real entries — their properties, minors and cofactors, and applications: the area of a triangle, the adjoint and inverse of a square matrix, the consistency of linear systems, and solving systems by the inverse-matrix method.

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