Problem

(i)

$\mathbf{A} = \begin{pmatrix} -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix}$

(a) Describe fully the single transformation represented by the matrix $\mathbf{A}$.

The matrix $\mathbf{B}$ represents a stretch scale factor 2 parallel to the $x$-axis.

(b) Write down the matrix $\mathbf{B}$.

The transformation represented by matrix $\mathbf{B}$ followed by the transformation represented by matrix $\mathbf{A}$ is the transformation represented by the matrix $\mathbf{C}$.

(c) Determine $\mathbf{C}$.

(ii)

$\mathbf{M} = \begin{pmatrix} k & -2 \\ -1 & 2k \end{pmatrix}$

where $k$ is a constant.

Given that the transformation represented by matrix $\mathbf{M}$ maps the point $(k, k)$ onto the point $(35, 91)$

(a) determine the value of $k$.

A quadrilateral $Q$ is transformed to another quadrilateral $Q'$ by the matrix $\mathbf{M}$.

Given that $Q'$ has area 336

(b) use the value of $k$ found in part (ii)(a) to determine the area of $Q$.