The matrix M is given by `M = beginpmatrix 1 & 0 k² & k endpmatrix`, where k is a constant and `k ≠ 0` or `1`.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

1. [B2] Shear [in the y-direction] (factor k) followed by a stretch, [parallel to the y-axis, scale factor k]. (Award B1 if given in the wrong order.)
2. [B1] `M^{-1} = \begin{pmatrix} 1 & 0 \\ -k & 1/k \end{pmatrix}`
3. [B1] `\begin{pmatrix} 1 & 0 \\ -k & 1/k \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -k & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1/k \end{pmatrix}` (Correct order.)
4. [B1] `\begin{pmatrix} 1 & 0 \\ k^2 & k \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}`
5. [M1] `k^2x + ky = y` (Sets `\begin{pmatrix} x \\ y \end{pmatrix}` for invariant points.)
6. [A1] `k^2x = (1-k)y`
7. [A1] `y = (k²)/(1-k)x` (AG.)
8. [M1] `|\det M| = 1` (Uses `Area(DEF) = |\det M| Area(ABC)`.)
9. [A1] `k = -1`