Problem

(a) Prove by induction that for $n \in \mathbb{Z}^+$

$\begin{pmatrix} 1 & r \\ 0 & 2 \end{pmatrix}^n = \begin{pmatrix} 1 & (2^n - 1)r \\ 0 & 2^n \end{pmatrix}$

where $r$ is a constant.

$\mathbf{M} = \begin{pmatrix} 4 & 0 \\ 0 & 5 \end{pmatrix} \quad \mathbf{N} = \begin{pmatrix} 1 & 2 \\ 0 & 2 \end{pmatrix}$

The transformation represented by matrix $\mathbf{M}$ followed by the transformation represented by matrix $\mathbf{N}$ is represented by the matrix $\mathbf{B}$

(b) (i) Determine $\mathbf{B}$ in the form $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ where $a$, $b$, $c$ and $d$ are integers.

(ii) Determine $\mathbf{B}^n$

Hexagon $S$ is transformed onto hexagon $S'$ by matrix $\mathbf{B}$

(c) Given that the area of $S'$ is $720$ square units, determine the area of $S$