Chapter 3 · Introduction

Chapter 3 — Determinants

One number computed from a square matrix answers two questions at once: is the matrix invertible, and by how much does it stretch area and volume?

Imagine you are handed a $10\times10$ matrix — the stiffness matrix of a small bridge model, say, or the coefficients of an economic model — and asked a simple yes-or-no question: does the system $A\mathbf{x}=\mathbf{b}$ have exactly one solution for every right-hand side? Chapter 2 gave you a way to answer: row reduce and count pivots. This chapter gives you something more compact. From the entries of any square matrix we can compute a single number, the determinant $\det A$, and the answer is "yes" exactly when $\det A\ne0$.

The same number has a geometric meaning that you can see. A $2\times2$ matrix moves the points of the plane; the unit square becomes a parallelogram, and $|\det A|$ is that parallelogram's area. Every region in the plane has its area multiplied by the same factor $|\det A|$, and in three dimensions $|\det A|$ multiplies volumes. When $\det A=0$ the plane is squashed flat onto a line — information is lost, and no inverse can bring it back. That is why the algebraic test and the geometric picture agree. Along the way we will meet formulas that express solutions and inverses as ratios of determinants, and use determinants to measure land plots, crystals and ellipses.

2×2: ad − bc→Minors & cofactor expansion→Row operations & fast computation→det A ≠ 0 ⇔ A invertible→det(AB) = det A · det B→Cramer's rule & adjugate→|det A| = area / volume scale

🎯 By the end of this chapter you can…

compute determinants of $2\times2$ and $3\times3$ matrices quickly, and of larger matrices by cofactor expansion along a well-chosen row or column;
find the determinant of a triangular matrix at a glance, and compute any determinant efficiently by row reduction, tracking interchanges and scalings;
use $\det A\ne0$ to decide invertibility, and apply the rules for $\det(AB)$, $\det A^{T}$, $\det A^{-1}$ and $\det(kA)$;
solve small systems with Cramer's rule and find inverses with the adjugate — and know when not to;
compute areas of parallelograms and triangles, volumes of parallelepipeds and tetrahedra, and the factor by which a linear map scales area or volume.

Watch a $2\times2$ matrix act on the plane. The grid of unit squares is carried along, and the shaded unit square becomes a parallelogram whose area is $|\det A|$. Every little square grows by the same factor. When $\det A=0$ the whole plane is squashed onto a line: the matrix cannot be undone. The orange marker flips over when $\det A\lt 0$. Press Play to cycle through the examples.

Matrix $A$:morph:

$\det A=ad-bc$—

area of the image of the unit square—

invertible?—

Sections in this chapter

3.1

Determinants by Cofactor Expansion

The $2\times2$ formula, minors and cofactors, expansion along any row or column, Sarrus for $3\times3$, and triangular matrices.

$\det A=\sum_{j}a_{ij}C_{ij}$

Open section →3.2

Properties of Determinants

How row operations change the determinant, computing it by row reduction, the invertibility test, and products, transposes and inverses.

$\det(AB)=\det A\,\det B$

Open section →3.3

Cramer's Rule, the Adjoint, Area and Volume

Solutions and inverses as ratios of determinants; areas, volumes and the determinant as a scaling factor.

$x_i=\dfrac{\det A_i(\mathbf{b})}{\det A}$

Open section →

✅ Before you start

You need to be fluent with the elementary row operations and row reduction to echelon form (1.2 Gaussian and Gauss–Jordan Elimination), with matrix multiplication and the transpose (2.1 Matrix Operations), and with inverses, elementary matrices and the $2\times2$ inverse formula (2.2 Inverses and Elementary Matrices). From school you need the areas of a parallelogram (base $\times$ height) and a triangle, and the volume of a box.

← 2.4 Markov Chains and Input–Output Models 3.1 Determinants by Cofactor Expansion →