- (a) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)
$$\left( \begin{array} { l l }
1 & r
0 & 2
\end{array} \right) ^ { n } = \left( \begin{array} { c c }
1 & \left( 2 ^ { n } - 1 \right) r
0 & 2 ^ { n }
\end{array} \right)$$
where \(r\) is a constant.
$$\mathbf { M } = \left( \begin{array} { l l }
4 & 0
0 & 5
\end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r }
1 & - 2
0 & 2
\end{array} \right) ^ { 4 }$$
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 \(\mathbf { N }\) in the form \(\left( \begin{array} { l l } a & b
c & d \end{array} \right)\) where \(a , b , c\) and \(d\) are integers.
(ii) Determine B
Hexagon \(S\) is transformed onto hexagon \(S ^ { \prime }\) by matrix \(\mathbf { B }\)
(c) Given that the area of \(S ^ { \prime }\) is 720 square units, determine the area of \(S\)