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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

LearnYFM01WFM01 · Further Pure Mathematics 1Transformations using matricesTransformations, invariant points and lines

Transformations using matrices · Subtopic 1 of 1

Matrix Transformations and Invariance

The columns of a transformation matrix are the images of the unit vectors $\mathbf i$ and $\mathbf j$ . Rotations, reflections, enlargements and stretches all have standard matrices. $\mathbf{BA}$ means "do $\mathbf A$ , then $\mathbf B$ ". Invariant points satisfy $\mathbf M\mathbf x=\mathbf x$ ; invariant lines map onto themselves.

Cambridge 9231 Paper 1Edexcel YFM01 WFM01Edexcel 9FM0 Paper 1Edexcel 9FM0 Paper 2AQA 7367 Paper 1AQA 7367 Paper 2≈ 45 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Find the matrix of a transformation from the images of $\mathbf i$ and $\mathbf j$ .
Use standard matrices for rotations, reflections, enlargements and stretches in 2D, and rotations about the axes in 3D.
Combine transformations, remembering $\mathbf{BA}$ means $\mathbf A$ first.
Find invariant points and invariant lines.

Pause and think

A reflection in the line $y=x$ swaps the coordinates of every point. Which points don't move?

Section 2 of 14

Core idea

Start hereWhere do the corners go?

Every rotation, reflection or stretch of the plane that keeps the origin fixed is completely decided by just two arrows: where $(1,0)$ goes and where $(0,1)$ goes. Write those two images side by side and you have the transformation's matrix.

The big idea

Columns of $\mathbf M$ = images of $\mathbf i$ , $\mathbf j$ ( $\mathbf k$ ). " $\mathbf A$ then $\mathbf B$ " is $\mathbf{BA}$ . Invariant points: $\mathbf M\mathbf x=\mathbf x$ . Invariant lines map onto themselves.

1. Building matrices from unit vectors

Why the columns are the images

$\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}a\\c\end{pmatrix}$
Picks out column 1.
$\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}b\\d\end{pmatrix}$
Picks out column 2.
$\mathbf M\begin{pmatrix}x\\y\end{pmatrix}=x\begin{pmatrix}a\\c\end{pmatrix}+y\begin{pmatrix}b\\d\end{pmatrix}$
Every vector is $x\mathbf i+y\mathbf j$ , so its image is fixed by these two.

The rotation matrix

$\mathbf i\mapsto(\cos\theta,\sin\theta)$
Rotate $(1,0)$ anticlockwise by $\theta$ round the unit circle.
$\mathbf j\mapsto(-\sin\theta,\cos\theta)$
$(0,1)$ starts $90^\circ$ further round.
$\mathbf R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$
Write them as columns.

\text{Reflection in }y=x\tan\theta:\ \begin{pmatrix}\cos2\theta&\sin2\theta\\\sin2\theta&-\cos2\theta\end{pmatrix}

$\theta$: angle of the mirror line with the $x$ -axis.
$\det=-1$: reflections reverse orientation.

Key takeaway3D rotations Rotation about the

z

-axis:

\begin{pmatrix}\cos\theta&-\sin\theta&0\\\sin\theta&\cos\theta&0\\0&0&1\end{pmatrix}

; the axis's own unit vector stays fixed. Similarly for the

x

- and

y

-axes.

2. Combining transformations

Why "

\mathbf A

then

\mathbf B

" is

\mathbf{BA}

$\mathbf x\ \xrightarrow{\ \mathbf A\ }\ \mathbf{Ax}$
First transformation.
$\mathbf{Ax}\ \xrightarrow{\ \mathbf B\ }\ \mathbf B(\mathbf{Ax})=(\mathbf{BA})\mathbf x$
Second acts on the result; associativity.

Worked exampleReflect then rotate

Reflection in the $x$ -axis, then rotation $90^\circ$ anticlockwise about $O$ . Single transformation?

Goal: Matrix and description.
Strategy: $\mathbf{BA}$ .

$\mathbf A=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\ \mathbf B=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$
Standard matrices.
$\mathbf{BA}=\begin{pmatrix}0&1\\1&0\end{pmatrix}$
Multiply in the right order.

AnswerReflection in

y=x

✓ Check

(1,0)\to(1,0)\to(0,1)

; reflection in

y=x

also sends

(1,0)

(0,1)

✓.

3. Invariant points and lines

Invariant lines through the origin

$\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\mx\end{pmatrix}=\begin{pmatrix}(a+bm)x\\(c+dm)x\end{pmatrix}$
Image of a general point on $y=mx$ .
$c+dm=m(a+bm)$
The image must also lie on $y=mx$ .
$bm^2+(a-d)m-c=0$
A quadratic in $m$ : up to two invariant lines through $O$ .

For lines not through

O

, use

y=mx+c

and compare both gradient and intercept.

Worked example $\mathbf M=\begin{pmatrix}3&1\\2&2\end{pmatrix}$

Find its invariant lines through $O$ .

Goal: Gradients.
Strategy: Quadratic above.

$m^2+(3-2)m-2=0$
$b=1$ , $a-d=1$ , $c=2$ .
$(m-1)(m+2)=0$
Factorise.

Answer

y=x

and

y=-2x

✓ Check

(1,1)\to(4,4)

y=x

✓;

(1,-2)\to(1,-2)

y=-2x

✓ (in fact a line of invariant points).

Common misconceptionInvariant line vs line of invariant points On an invariant line, points may slide along it; on a line of invariant points, every point stays put.

Pause and think

Which transformation has every line through $O$ invariant?

Summary

Columns = images of unit vectors.
$\mathbf{BA}$ for " $\mathbf A$ then $\mathbf B$ ".
$\det$ : area factor; negative = reflection-type.
Invariant lines from a quadratic in $m$ .

Linear transformation: Map preserving addition and scaling; fixes the origin.
Invariant line: Line mapped onto itself.

HistoryLinear algebra. Hermann Grassmann's 1844 work on "extension" treated geometry with linear transformations. It was so ahead of its time that it was largely ignored for decades.

Fun factPhone screens. Rotating your phone triggers a

90^\circ

rotation matrix applied to every element on screen.

Section 3 of 14

Key formulae

Rotation by θ anticlockwise

\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}

Reflection in y = (tan θ)x

\begin{pmatrix}\cos2\theta&\sin2\theta\\\sin2\theta&-\cos2\theta\end{pmatrix}

Enlargement / stretch

\begin{pmatrix}k&0\\0&k\end{pmatrix},\qquad\begin{pmatrix}k&0\\0&1\end{pmatrix}\ (x\text{-direction})

Invariance

\mathbf M\mathbf x=\mathbf x\ (\text{points});\qquad y=mx+c\mapsto y=mx+c\ (\text{lines})

TrickWhere do i and j go? Sketch the unit square, draw where

(1,0)

and

(0,1)

end up, and write those two vectors as columns. That's the matrix, every time.

Section 4 of 14

Worked examples

Example 1 · Combined transformation0 / 3 steps

\mathbf A

: reflection in the

x

-axis.

\mathbf B

: rotation

90^\circ

anticlockwise about

O

. Find the single matrix for "

\mathbf A

then

\mathbf B

Example 2 · Invariant lines0 / 3 steps

Find the invariant lines through the origin of

\mathbf M=\begin{pmatrix}3&1\\2&2\end{pmatrix}

Section 5 of 14

Explore the graph

Rotation matrix

Loading the interactive graph…

Try this

The two points are the images of $\mathbf i$ and $\mathbf j$ : the columns of the rotation matrix.
Set $t=\frac\pi2$ : $\mathbf i\to(0,1)$ , $\mathbf j\to(-1,0)$ .

Columns $(\cos t,\sin t)$ and $(-\sin t,\cos t)$ .

In the real worldComputer graphics. Graphics cards chain transformations by multiplying matrices once, then apply the single product to every vertex, which is far faster than applying each transformation separately.

Section 6 of 14

Common mistakes

Incorrect"

\mathbf A

then

\mathbf B

" is

\mathbf{AB}

Correct

\mathbf{BA}

The matrix nearest the vector acts first.

IncorrectInvariant line = line of invariant points

CorrectPoints may move along an invariant line

Only the line as a whole must map to itself.

IncorrectRotation clockwise with the anticlockwise matrix

CorrectUse

-\theta

for clockwise

The standard matrix is anticlockwise.

Exam: "A followed by B"

Me, wisely: so that's BA.

Section 7 of 14

Exam tip

Exam tipDescribe fully. "Rotation" alone scores nothing. Give the angle, the direction and the centre (or axis in 3D).

Section 8 of 14

Guided practice

SA-FP-MTR-G011 markL1 FoundationNon-calculatorStandard matrices

The matrix

\begin{pmatrix}3&0\\0&3\end{pmatrix}

is an enlargement about

O

. Find its area scale factor.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FP-MTR-0012 marksL2 StandardNon-calculatorStandard matrices

The matrix

\begin{pmatrix}\frac12&-\frac{\sqrt3}2\\\frac{\sqrt3}2&\frac12\end{pmatrix}

is a rotation about

O

. Find the anticlockwise angle in degrees (

0\le\theta<360

Salam Academy Original Practice · Proposed mark scheme

SA-FP-MTR-0022 marksL2 StandardNon-calculatorCombining

\mathbf P=\begin{pmatrix}2&0\\0&1\end{pmatrix}

(stretch) and

\mathbf Q=\begin{pmatrix}0&-1\\1&0\end{pmatrix}

(rotation). The point

(1,3)

undergoes

\mathbf P

then

\mathbf Q

. Find the

x

-coordinate of its image.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-MTR-0033 marksL3 ExamNon-calculatorInvariance

Find the gradients of the invariant lines through

O

\begin{pmatrix}3&1\\2&2\end{pmatrix}

Salam Academy Original Practice · Proposed mark scheme

SA-FP-MTR-0043 marksL3 ExamNon-calculatorInvariance

\mathbf M=\begin{pmatrix}2&1\\1&2\end{pmatrix}

has a line of invariant points

y=kx

. Find

k

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-FP-MTR-0053 marksL4 ChallengingNon-calculatorStandard matrices

The matrix

\begin{pmatrix}0&1\\1&0\end{pmatrix}

is a reflection in

y=(\tan\theta)x

. Find

\theta

in degrees (

0\le\theta<180

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Columns = images of $\mathbf i$ , $\mathbf j$ (and $\mathbf k$ ).
$\mathbf{BA}$ : $\mathbf A$ first.
Invariant points: $\mathbf M\mathbf x=\mathbf x$ .
Invariant lines: substitute $y=mx(+c)$ .

Section 12 of 14

Quick quiz

Q1 · SA-FP-MTR-Q011 markL1 FoundationNon-calculatorCombining

Transformation

\mathbf A

is followed by

\mathbf B

. The single matrix is

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Further Mathematics (YFM01) and A level Further Mathematics (9FM0)
Where: Core Pure · Matrices (IAL: WFM01 Transformations using matrices)
Expected: Linear transformations in 2D and 3D; successive transformations; invariant points and lines; area scale factor.
Calculator: Permitted; show enough working to earn the method marks.
Typical style: Edexcel often opens with a routine part and builds to a "show that" or "prove" part.
Open in: YFM01 · WFM01 9FM0 · Paper 1 9FM0 · Paper 2

Section 14 of 14

Past paper questions

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YFM01, 2018 specification (official specification).