Key Idea: A 2×2 matrix can move a shape: rotate it, enlarge it or reflect it. Its determinant tells you how the area changes and whether the shape turns over.
Transformation matrices
- Image = matrix × column vector, with the matrix on the left.
- Rotation (anticlockwise about O): $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$. Reflection in y = (tan θ)x: $\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}$.
- Enlargement factor k, centre O: $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$
- For a shape, transform each corner, then join them again.
| Idea | Rule |
|---|---|
| A then B | Multiply as BA; the right one acts first |
| Determinant | ad − bc |
| New area | |det| × old area |
| det < 0 | The shape is flipped over |
| det = 0 | It collapses; no inverse |
| Product | det(BA) = det B · det A |
When you combine matrices, track one point to check which one acts first.
A triangle with corners (1, 0), (2, 0), (2, 1) is enlarged by factor 2 about O. Find the images. (2, 0), (4, 0), (4, 2).
What happens to that triangle's area? It is multiplied by 4, since det = 2 × 2 = 4.
A logo of area 9 is enlarged by 2, then rotated (det R = 1). New area? 4 × 9 = 36.
A matrix has det = −3. What does it do to a shape? It triples the area and flips the shape over.
How do you rotate clockwise? Use −θ in the rotation matrix.