Reflect in x-axis
100−1
🎯 Learning Objectives
- use 2×2 matrices to transform coordinates;
- recognise standard reflection, rotation and enlargement matrices;
- determine a matrix from basis-vector images;
- interpret a transformation matrix.
Introduction
This lesson develops Transformation Matrices from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Matrix Mapping
If A is a transformation matrix, then image vector = A × object vector.
The first column is the image of (1,0); the second column is the image of (0,1).
Standard Matrices
Reflect in y-axis
−1001
Reflect in y=x
0110
90° anticlockwise
0−110
180° rotation
−100−1
Enlargement k
k00k
Worked Example
Apply 0−110 to (3,2).
x′=0(3)−1(2)=−2.
y′=1(3)+0(2)=3.
Image=(−2,3), a 90° anticlockwise rotation.
Practice Questions
- Apply 100−1 to (4,−3).
- Apply 2002 to (−1,5).
- Identify 0110.
✅ Check Your Work
Show answers and working
- (4,3), reflection in x-axis.
- (−2,10), enlargement scale factor 2 about origin.
- Reflection in y=x.
🧠 Learn It Visually: Transformation Matrices
A 2×2 matrix can act like a movement rule. Multiplying the matrix by a point’s column vector produces the image coordinates.
Point as a column vector
Represent (x,y) as a vertical vector with x above y.
Multiply carefully
Use row-by-column multiplication to find x′ and y′.
Recognise common matrices
Different matrices represent reflections, rotations, stretches, enlargements and shears.
🎛️ See the Transformation Happen
Select an option and watch the object change. Compare orientation, size and position.
Choose a transformation.
Think Like a Mathematician: Why must every vertex be multiplied by the same transformation matrix?
🧷 Remember
The matrix acts on position vectors measured from the origin, so these transformations are centred on or fixed relative to the origin.
⚠️ Common Mistake
Multiplying coordinates in the wrong order or treating the point as a row vector.
🎯 Exam Tip
Calculate one vertex, check the expected movement, then transform the remaining vertices.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.