How to find a transformation matrix
Read a transformation matrix straight off the images of the unit vectors, with a worked example and the order rule for combined transformations.
What the examiner is testing
Whether you can move between a description of a transformation and its matrix, in both directions, and whether you can combine two of them in the right order.
The unit vector method
Any transformation matrix is built from where the two unit vectors land:
$$ \mathbf{M} = \begin{pmatrix} \text{image of } (1,0) & \text{image of } (0,1) \end{pmatrix} $$
Each image is written as a column. That is the whole trick, and it works for every linear transformation on the syllabus.
Worth memorising:
| Transformation | Image of (1,0) | Image of (0,1) | Matrix |
|---|---|---|---|
| Reflection in the x-axis | (1, 0) | (0, −1) | \( \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \) |
| Reflection in the y-axis | (−1, 0) | (0, 1) | \( \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \) |
| Reflection in \( y = x \) | (0, 1) | (1, 0) | \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \) |
| Rotation 90° anticlockwise about O | (0, 1) | (−1, 0) | \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \) |
| Enlargement, scale factor \( k \) | (k, 0) | (0, k) | \( \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \) |
Worked example
Triangle \( ABC \) has vertices \( A(1, 1) \), \( B(4, 1) \), \( C(1, 3) \). It is rotated 90° anticlockwise about the origin to give \( A'B'C' \). Find the matrix and the image coordinates.
Step 1 — build the matrix. Under this rotation \( (1,0) \to (0,1) \) and \( (0,1) \to (-1,0) \). As columns:
$$ \mathbf{M} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} $$
Step 2 — apply it to all three vertices at once. Put the coordinates in as columns:
$$ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 4 & 1 \\ 1 & 1 & 3 \end{pmatrix} = \begin{pmatrix} -1 & -1 & -3 \\ 1 & 4 & 1 \end{pmatrix} $$
Answer: \( A'(-1, 1) \), \( B'(-1, 4) \), \( C'(-3, 1) \).
Check: the determinant is \( (0)(0) - (-1)(1) = 1 \), so area is unchanged — correct for a rotation. Any rotation or reflection must give a determinant of \( \pm 1 \).
Combined transformations
For transformation \( A \) followed by \( B \), the single matrix is \( \mathbf{BA} \) — not \( \mathbf{AB} \). The first transformation sits nearest the coordinates it acts on. Matrix multiplication is not commutative, so the wrong order gives a genuinely different answer.
The three mistakes that lose marks
1. Writing the images as rows. They are columns. A transposed matrix reflects when it should rotate.
2. Multiplying combined transformations in reading order. First applied goes on the right.
3. Forgetting that rotation is about the origin unless stated otherwise. If the centre is elsewhere, translate to the origin, rotate, translate back.
30-second recap
Track \( (1,0) \) and \( (0,1) \), write their images as columns. Apply the matrix to all vertices in one multiplication. Check the determinant: \( \pm 1 \) for rotations and reflections, \( k^2 \) for an enlargement. Combined transformations multiply right to left.