6 Let \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0
1 & 1 \end{array} \right)\).
- 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 \(d\).
- Prove by mathematical induction that, for all positive integers \(n\),
$$\mathbf { A } ^ { n } = \left( \begin{array} { c c }
2 ^ { n } & 0
2 ^ { n } - 1 & 1
\end{array} \right)$$ - 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 \(n\).