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AS Level · Further Pure Mathematics 1 WFM01

Matrix Algebra

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Edexcel IAL · Further Pure 1

Matrix Algebra

Matrices are rectangular grids of numbers with special multiplication rules — master those rules and you can manipulate, invert, and prove relationships between them.

Chapter Overview

What's in this chapter?

Four big topics — understand each one fully before moving to the next.

Introduction: what a matrix is, types, notation, and basic operations (scalar multiply, add, subtract)

Multiplying matrices: row × column rule, squaring, general dimensions, NOT commutative but IS associative

Inverse matrices: determinant, singular vs non-singular, formula for 2×2 inverse, (AB)⁻¹ = B⁻¹A⁻¹

Proving matrix relationships: pre- and post-multiplying, making a matrix the subject of an equation

Topic 1

Introduction to Matrices

What is a matrix?

Think of a matrix as a table of numbers arranged in rows (going across) and columns (going down). The order of a matrix is always written as rows × columns — number of rows first, columns second.

Key Notation

Order = rows × columns Element a_{i,j} = element in row i, column j A = [ a b ] is a 2 × 2 matrix [ c d ] B = [ a ] is a 2 × 1 matrix (column vector) [ b ]

💡 Memory trick: "Rows Before Columns" — just like how you read (left to right, then top to bottom). An m×n matrix has m rows and n columns.

Types of Matrix — know all five

// Column Matrix (m × 1) Row Matrix (1 × n) Square Matrix (n × n) [ a ] [ a b c ] [ a b ] [ b ] [ c d ] [ c ] Zero Matrix (0) Identity Matrix (I) [ 0 0 ] [ 1 0 ] [ 0 0 ] [ 0 1 ] All entries = 0 Diagonal = 1, rest = 0

✅ Identity matrix I acts like the number 1 in regular arithmetic. Multiplying any (compatible) matrix by I leaves it unchanged:

AI = IA = A

Scalar Multiplication

To multiply a matrix by a scalar (a number), simply multiply every single element by that number.

Worked Example — Scalar Multiply

Find 3A where A = [[2, −1], [4, 0]]

1Multiply every element by 3:

3A = 3 × [ 2 -1 ] = [ 3×2 3×(-1) ] = [ 6 -3 ] [ 4 0 ] [ 3×4 3×0 ] [ 12 0 ]

Adding and Subtracting Matrices

You can only add or subtract two matrices if they have the same order. Just add or subtract the corresponding elements.

⚠️ Common mistake: Students try to add matrices of different sizes. You cannot add a 2×3 matrix to a 3×2 matrix — the orders must match exactly.

Properties of Matrix Addition

Property	Addition	Subtraction
Commutative? (swap order OK?)	✓ YES — A + B = B + A	✗ NO — A − B ≠ B − A
Associative? (regroup OK?)	✓ YES — (A+B)+C = A+(B+C)	✗ NO — (A−B)−C ≠ A−(B−C)
Zero element	A + 0 = A (zero matrix has no effect)

Practice Question 1

Given A = [[3, −2], [1, 5], [0, 4]] and B = [[−1, 6], [3, −3], [2, 1]], find A + B and −2B.

Topic 2

Multiplying Matrices

The Golden Rule — when can you multiply?

To multiply matrix A by matrix B, the number of columns in A must equal the number of rows in B. Think of it as a "middle number" check.

Dimension Rule

Topic 3

Inverse Matrices

Topic 4

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Also in the full note

Proving Matrix Relationships
What to Memorise
Can you do all of these?
What Examiners Look For
How to actually multiply — the Row × Column method
Squaring a Matrix
Matrix Multiplication is NOT Commutative
Matrix Multiplication IS Associative

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Matrix Algebra

Matrix Algebra

What's in this chapter?

Introduction to Matrices

What is a matrix?

Types of Matrix — know all five

Scalar Multiplication

Adding and Subtracting Matrices

Properties of Matrix Addition

Multiplying Matrices

The Golden Rule — when can you multiply?

Inverse Matrices

More Further Pure Mathematics 1 topics