Chapter 2 · Introduction

Chapter 2 — Matrices

Matrices stop being bookkeeping for linear systems and become objects of their own: we add them, multiply them, undo them, factorise them — and use them to model how systems change.

In Chapter 1 a matrix was a tidy way of writing down a linear system so that we could row reduce it. In this chapter matrices become objects in their own right, with an algebra of their own. Every frame of a video game or an animated film is produced by multiplying the coordinates of millions of points by small matrices that rotate, stretch and project the scene; your phone does the same when it straightens a photo. A spreadsheet of sales multiplied by a table of prices gives revenue at a stroke. Weather models, search engines and epidemic forecasts all repeat one matrix multiplication over and over to see where a system is heading.

To work with matrices we need to know how to add and multiply them — and multiplication turns out to be stranger than for numbers, because the order matters. We then ask when a matrix can be undone: that is the inverse, and the row operations of Chapter 1 reappear as multiplications by elementary matrices. Engineers solving the same system for thousands of right-hand sides go one step further and factorise the matrix as $A=LU$, so each new solve is quick. Finally, two classic models — Markov chains, which follow how a population moves between states, and Leontief's input–output model of an economy, which earned a Nobel Prize — show this algebra at work.

Tables of numbers→Matrix algebra: $A+B$, $cA$, $AB$, $A^{T}$→Undoing: $A^{-1}$ and elementary matrices→Factorising: $A=LU$→Modelling: Markov chains, input–output
🎯 By the end of this chapter you can…
  • add, scale, multiply and transpose matrices, decide when a product is defined, and read $A\mathbf{x}$ as a linear combination of the columns of $A$;
  • recognise the ways matrix algebra differs from ordinary algebra: $AB\ne BA$, $AB=O$ with $A,B\ne O$, no cancelling;
  • decide whether a square matrix is invertible, compute $A^{-1}$ by the $2\times2$ formula or by row reducing $\left[\begin{array}{c|c}A&I\end{array}\right]$, and use it to solve $A\mathbf{x}=\mathbf{b}$;
  • describe row operations as multiplication by elementary matrices and use the first version of the Invertible Matrix Theorem;
  • factorise a matrix as $A=LU$ and use the factorisation to solve systems efficiently;
  • build and run Markov-chain and input–output models, and interpret their long-run behaviour.
Teaser — Does the Order Matter?

Each matrix moves the points of the plane. On the left the letter F is moved first by $A$ and then by $B$ — the combined matrix is the product $BA$. On the right the order is reversed, giving $AB$. Press Play and compare the final pictures for each pair.

first A, then B: the matrix $BA$
first B, then A: the matrix $AB$
$A$—
$B$—
$BA$—
$AB$—

Sections in this chapter

2.1

Matrix Operations

Sums, scalar multiples and the row-by-column product; $A\mathbf{x}$ as a combination of columns; transposes and symmetric matrices.

$(AB)_{ij}=\sum_k a_{ik}b_{kj}$
Open section →2.2

Inverses and Elementary Matrices

Undoing a matrix, row operations as multiplications, $\left[\begin{array}{c|c}A&I\end{array}\right]\to\left[\begin{array}{c|c}I&A^{-1}\end{array}\right]$, and the Invertible Matrix Theorem.

$A^{-1}=\dfrac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$
Open section →2.3

LU Factorisation

Record elimination as a lower-triangular $L$ and an echelon $U$, then solve $A\mathbf{x}=\mathbf{b}$ by two quick triangular substitutions; row swaps give $PA=LU$, and operation counts show why it pays.

$A=LU$
Open section →2.4

Markov Chains and Input–Output Models

Populations moving between states step by step, steady states, and how much each sector of an economy must produce.

$\mathbf{x}_{k+1}=P\mathbf{x}_k$
Open section →
✅ Before you start
You need the row reduction of Chapter 1: writing a system as an augmented matrix and solving it (1.1 Introduction to Linear Systems), elementary row operations, echelon and reduced echelon form, pivots and free variables, and deciding whether a system has no, one or infinitely many solutions (1.2 Gaussian and Gauss–Jordan Elimination). The models of 1.3 Applications (traffic flow, mixtures, networks) are a good warm-up for Section 2.4. From school you need confident arithmetic with fractions and negative numbers, and coordinates in the plane — the widgets draw matrices acting on points $(x,y)$.
← 1.3 Applications of Linear Systems2.1 Matrix Operations →