6 Let $\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 1 & 1 \end{array} \right)$.\\
(a) The transformation in the $x - y$ plane represented by $\mathbf { A } ^ { - 1 }$ transforms a triangle of area $30 \mathrm {~cm} ^ { 2 }$ into a triangle of area $d \mathrm {~cm} ^ { 2 }$.

Find the value of $d$.\\

(b) Prove by mathematical induction that, for all positive integers $n$,

$$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 
2 ^ { n } & 0 \\
2 ^ { n } - 1 & 1
\end{array} \right)$$

(c) The line $y = 2 x$ is invariant under the transformation in the $x - y$ plane represented by $\mathbf { A } ^ { n } \mathbf { B }$, where $\mathbf { B } = \left( \begin{array} { r l } 1 & 0 \\ 33 & 0 \end{array} \right)$.

Find the value of $n$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q6}}