Chapter 3 · Introduction

Chapter 3 — Matrices

A matrix is a rectangular table of numbers that can be added, multiplied and sometimes inverted — the language behind spreadsheets, computer graphics and systems of equations.

Open any spreadsheet and you are looking at a matrix: a rectangular block of numbers arranged in rows and columns. A shop's weekly sales of three items in four branches, the marks of a class in five subjects, the brightness of every pixel in a phone photograph, the road distances between six cities — all of these are tables, and in this chapter tables become mathematical objects. We give each table a name such as $A$, describe its shape by its order $m\times n$, and learn when two tables are equal, how to add them and how to multiply one by a number.

The surprise comes with multiplication of two matrices. It is defined in a “row times column” way that looks strange at first, yet it is exactly what you need to turn quantities and prices into bills, or to apply one geometric transformation after another. This product breaks two rules you have trusted since primary school: $AB$ need not equal $BA$, and $AB$ can be the zero matrix even when neither $A$ nor $B$ is zero. We then flip matrices over their diagonal (the transpose), split every square matrix into a symmetric and a skew-symmetric part, and ask when a matrix can be “undone” by an inverse. Chapter 4 (Determinants) completes the story by giving a test and a formula for the inverse.

Tables of numbers→Order & types→Add · scale · multiply→$AB\ne BA$→Transpose & symmetry→Inverse $A^{-1}$→Determinants (Ch 4)
🎯 By the end of this chapter you can…
  • write data as a matrix, state its order, read any entry $a_{ij}$, and construct a matrix from a rule such as $a_{ij}=\tfrac12(i+2j)^2$;
  • recognise row, column, square, diagonal, scalar, identity and zero matrices, and use equality of matrices to find unknown entries;
  • add, subtract and scale matrices, multiply two matrices when their orders allow it, and use the properties of these operations — while showing by example that $AB\ne BA$ in general and that $AB=O$ does not force $A=O$ or $B=O$;
  • find transposes, use $(AB)'=B'A'$, classify symmetric and skew-symmetric matrices, and write any square matrix as the sum of a symmetric and a skew-symmetric matrix;
  • verify that one matrix is the inverse of another, prove that an inverse is unique and that $(AB)^{-1}=B^{-1}A^{-1}$, and find the inverse of a small matrix by solving equations;
  • answer the Board's 1-, 2-, 3- and 5-mark questions and the JEE-style MCQs on matrices.
Teaser — A 2 × 2 Matrix Moves the Plane

The matrix $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ sends each point $(x,y)$ to $(ax+by,\ cx+dy)$. Watch what it does to the unit square (dashed) and the little flag inside it: the corner $(1,0)$ goes to the first column $(a,c)$ and $(0,1)$ goes to the second column $(b,d)$. Pick a preset or move the sliders; Play morphs the square from “do nothing” ($I$) to $A$ and back.

matrix A—
(1, 0) ↦ first column—
(0, 1) ↦ second column—
area scale factor ad − bc—

The sliders move in steps of $\tfrac12$. Keep an eye on the last readout: when $ad-bc=0$ the square is flattened onto a line and no matrix can bring it back — that is the first hint of the inverse matrices of Section 3.4 and the determinants of Chapter 4.

Sections in this chapter

3.1

Matrices and Types of Matrices

Matrices as tables, order and entries $a_{ij}$; row, column, square, diagonal, scalar, identity and zero matrices; equality of matrices.

$A=[a_{ij}]_{m\times n}$
Open section →3.2

Operations on Matrices

Addition, scalar multiples and the row-by-column product, with their properties; why $AB\ne BA$ and how $AB$ can be $O$.

$(AB)_{ik}=\sum_j a_{ij}b_{jk}$
Open section →3.3

Transpose, Symmetric and Skew-Symmetric Matrices

The transpose and its properties, including the reversal law; symmetric and skew-symmetric matrices; splitting any square matrix into the two.

$(AB)'=B'A'$
Open section →3.4

Invertible Matrices

When a square matrix can be undone; uniqueness of the inverse; the inverse of a product; finding a small inverse by solving equations.

$AB=BA=I\ \Rightarrow\ B=A^{-1}$
Open section →
✅ Before you start
You need arithmetic with negative numbers and fractions, and solving pairs of linear equations in two unknowns (Class 10 Mathematics, Ch. 3 — Pair of Linear Equations in Two Variables) or three unknowns. The idea of a function and of a bijection from §1.2 Types of Functions returns in Section 3.4: an invertible matrix is exactly a matrix whose “action” on the plane can be run backwards. Coordinates of points and the area of simple figures (Class 11 Mathematics, Ch. 9 — Straight Lines) help with the pictures. Quick self-test: solve $2x+y=5,\ x-y=1$ — you should get $x=2$, $y=1$.
← 2.2 Graphs and Principal Values3.1 Matrices and Types of Matrices →