\(\mathbf { M } = \left( \begin{array} { l l } 3 & a 0 & 1 \end{array} \right) \quad\) where \(a\) is a constant
Prove by mathematical induction that, for \(n \in \mathbb { N }\)
$$\mathbf { M } ^ { n } = \left( \begin{array} { c c }
3 ^ { n } & \frac { a } { 2 } \left( 3 ^ { n } - 1 \right)
0 & 1
\end{array} \right)$$
Triangle \(T\) has vertices \(A , B\) and \(C\).
Triangle \(T\) is transformed to triangle \(T ^ { \prime }\) by the transformation represented by \(\mathbf { M } ^ { n }\) where \(n \in \mathbb { N }\)
Given that
triangle \(T\) has an area of \(5 \mathrm {~cm} ^ { 2 }\)
triangle \(T ^ { \prime }\) has an area of \(1215 \mathrm {~cm} ^ { 2 }\)