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Standard transformation matrices

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

F1 uses four families of transformation: reflections in the axes and in y = ±x, rotations about O, stretches parallel to the axes, and enlargements about O. You need to write down their matrices and recognise them.

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Reflections

In the x-axis: (1 0; 0 -1)
In the y-axis: (-1 0; 0 1)
In y = x: (0 1; 1 0)
In y = -x: (0 -1; -1 0)

Each one is easy to check: see where (1, 0) and (0, 1) go.

Rotations about O

Anticlockwise through θ: (cos θ -sin θ; sin θ cos θ). A clockwise rotation through θ uses -θ.

Special cases: 90° is (0 -1; 1 0), 180° is (-1 0; 0 -1). Use exact values: cos 60° = 1/2, sin 60° = sqrt(3)/2, cos 45° = sin 45° = sqrt(2)/2.

Stretches and enlargements

Stretch parallel to the x-axis, scale factor k: (k 0; 0 1).
Stretch parallel to the y-axis, scale factor k: (1 0; 0 k).
Enlargement about O, scale factor k (k ≠ 0): (k 0; 0 k). A negative k also turns the shape through 180°.

When describing a transformation, give every detail: the centre of a rotation or enlargement, the angle and direction, the mirror line, or the scale factor and direction of a stretch.

Worked example

Find the image of (4, 0) under a rotation of 60° anticlockwise about O.

Matrix (1/2 -sqrt(3)/2; sqrt(3)/2 1/2).
Image: (4 × 1/2, 4 × sqrt(3)/2).

Answer: (2, 2sqrt(3))

Key idea: Reflections, rotations (cos θ -sin θ; sin θ cos θ), stretches (k 0; 0 1) or (1 0; 0 k), enlargements (k 0; 0 k). Describe a transformation fully.

Check yourself

Which transformation does (0 -1; -1 0) represent?
- Reflection in y = x
- Rotation 90° anticlockwise about O
- Reflection in y = -x
- Rotation 180° about O
Show answer
Reflection in y = -x. (1, 0) goes to (0, -1) and (0, 1) goes to (-1, 0): the points swap and change sign, which is reflection in y = -x.
Find the y-coordinate of the image of (4, 0) under rotation 60° anticlockwise about O, to 3 decimal places.
Show answer
3.464. y = 4 sin 60° = 2sqrt(3) = 3.464.
Which matrix represents rotation 90° clockwise about O?
- (0 -1; 1 0)
- (-1 0; 0 -1)
- (0 1; -1 0)
- (0 1; 1 0)
Show answer
(0 1; -1 0). Clockwise 90° is θ = -90°: (cos(-90°) -sin(-90°); sin(-90°) cos(-90°)) = (0 1; -1 0). Check: (0, 1) goes to (1, 0).
(0.8 -0.6; 0.6 0.8) represents an anticlockwise rotation about O through θ°, with 0 < θ < 90. Find θ to 1 decimal place.
Show answer
36.9 °. cos θ = 0.8 and sin θ = 0.6, so θ = 36.9°.
Describe the transformation (1 0; 0 3).
- Stretch parallel to the x-axis, scale factor 3
- Enlargement scale factor 3 about O
- Rotation about O
- Stretch parallel to the y-axis, scale factor 3
Show answer
Stretch parallel to the y-axis, scale factor 3. x stays the same and y is multiplied by 3.
Describe the transformation (-2 0; 0 -2).
- Reflection in y = -x
- Enlargement scale factor -2 about O
- Stretch parallel to the x-axis, scale factor -2
- Rotation 180° about O
Show answer
Enlargement scale factor -2 about O. Both coordinates are multiplied by -2.
Find the x-coordinate of the image of (3, -5) after reflection in the line y = x.
Show answer
-5. (0 1; 1 0)(3, -5) = (-5, 3).

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Last updated 8 November 2026. Spotted a mistake? Tell us.