Problem

$\mathbf{A} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}$

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

The matrix $\mathbf{B}$ represents a rotation of $210°$ anticlockwise about centre $(0, 0)$.

(b) Write down the matrix $\mathbf{B}$, giving each element in exact form.

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

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

The hexagon $H$ is transformed onto the hexagon $H'$ by the matrix $\mathbf{C}$.

(d) Given that the area of hexagon $H$ is $5$ square units, determine the area of hexagon $H'$.