Why Matrices and Transformations Together?
On the IGCSE Extended paper, matrices are used to describe geometric transformations. Instead of describing a reflection in words, you can represent it as a 2×2 matrix that, when multiplied by a position vector, produces the image point. This is elegant and efficient — and it is a favourite topic for examiners.
You need to be able to work in both directions: given a matrix, identify the transformation, and given a transformation, write down the matrix.
The Key Transformation Matrices
Here are the standard 2×2 matrices you must know. Each acts on a column vector (x, y) by matrix multiplication.
Reflections
| Transformation | Matrix |
|---|---|
| Reflection in the x-axis | [[1, 0], [0, −1]] |
| Reflection in the y-axis | [[−1, 0], [0, 1]] |
| Reflection in y = x | [[0, 1], [1, 0]] |
| Reflection in y = −x | [[0, −1], [−1, 0]] |
Rotations (about the origin)
| Transformation | Matrix |
|---|---|
| 90° anticlockwise | [[0, −1], [1, 0]] |
| 90° clockwise | [[0, 1], [−1, 0]] |
| 180° | [[−1, 0], [0, −1]] |
Enlargements (centre origin)
An enlargement with scale factor k centred at the origin is represented by [[k, 0], [0, k]]. For example, scale factor 2 gives [[2, 0], [0, 2]], and scale factor −1 gives [[−1, 0], [0, −1]] (which is the same as a 180° rotation).
Shears and Stretches
A shear parallel to the x-axis with shear factor k is [[1, k], [0, 1]]. A stretch parallel to the y-axis with scale factor k is [[1, 0], [0, k]].
How to Apply a Transformation Matrix
To find the image of a point under a transformation, multiply the matrix by the position vector of the point.
Worked Example 1: Find the image of the point (3, 5) under reflection in the x-axis.
The matrix for reflection in the x-axis is [[1, 0], [0, −1]].
Multiply: [[1, 0], [0, −1]] × [[3], [5]] = [[1×3 + 0×5], [0×3 + (−1)×5]] = [[3], [−5]]
The image is (3, −5).
This makes geometric sense — reflecting in the x-axis keeps the x-coordinate the same and negates the y-coordinate.
Worked Example 2: Find the image of (2, −1) under a 90° anticlockwise rotation about the origin.
Matrix: [[0, −1], [1, 0]]
[[0, −1], [1, 0]] × [[2], [−1]] = [[0×2 + (−1)(−1)], [1×2 + 0×(−1)]] = [[1], [2]]
The image is (1, 2).
Identifying a Transformation from Its Matrix
This is a common exam question. You are given a matrix and asked to describe the single transformation it represents.
Strategy: Use the Unit Square
Apply the matrix to the standard basis vectors (1, 0) and (0, 1). The results tell you where the unit vectors map to, which reveals the transformation.
Worked Example 3: Describe the transformation represented by [[0, 1], [1, 0]].
Apply to (1, 0): [[0, 1], [1, 0]] × [[1], [0]] = [[0], [1]]
Apply to (0, 1): [[0, 1], [1, 0]] × [[0], [1]] = [[1], [0]]
The point (1, 0) maps to (0, 1) and (0, 1) maps to (1, 0). The x and y coordinates have been swapped. This is a reflection in the line y = x.
Worked Example 4: Describe the transformation represented by [[0, 1], [−1, 0]].
Apply to (1, 0): result is (0, −1). Apply to (0, 1): result is (1, 0).
The point (1, 0) has moved to (0, −1), which is a 90° clockwise rotation. The point (0, 1) has moved to (1, 0), confirming this. The transformation is a rotation of 90° clockwise about the origin.
Combined Transformations
When two transformations are applied in sequence, the combined effect is found by multiplying their matrices. Order matters — matrix multiplication is not commutative.
If transformation A is applied first and then transformation B, the combined matrix is BA (B times A, reading right to left).
Worked Example 5: A shape is first reflected in the y-axis, then rotated 90° anticlockwise about the origin. Find the single matrix representing the combined transformation.
Reflection in y-axis: M₁ = [[−1, 0], [0, 1]]
Rotation 90° anticlockwise: M₂ = [[0, −1], [1, 0]]
Combined matrix = M₂ × M₁ = [[0×(−1) + (−1)×0, 0×0 + (−1)×1], [1×(−1) + 0×0, 1×0 + 0×1]] = [[0, −1], [−1, 0]]
This is the matrix for reflection in the line y = −x. So the two transformations together are equivalent to a single reflection.
Inverse Transformations
The inverse of a transformation matrix undoes the transformation. For a 2×2 matrix [[a, b], [c, d]], the inverse is (1/det) × [[d, −b], [−c, a]], where det = ad − bc.
If a matrix represents a reflection, its inverse is itself (reflecting twice returns to the original). If a matrix represents a 90° clockwise rotation, its inverse represents a 90° anticlockwise rotation.
Common Mistakes
- Multiplying in the wrong order for combined transformations. Always apply the first transformation’s matrix on the right.
- Confusing clockwise and anticlockwise rotations. Check by applying the matrix to (1, 0).
- Forgetting that the transformations must be centred at the origin when using these standard matrices.
- Arithmetic errors in matrix multiplication. Write out each element carefully.
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