12 The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left[ \begin{array} { l l }
1 & 2
0 & 3
\end{array} \right]$$
12
- Prove by induction that, for all integers \(n \geq 1\),
$$\mathbf { A } ^ { n } = \left[ \begin{array} { c c }
1 & 3 ^ { n } - 1
0 & 3 ^ { n }
\end{array} \right]$$
12 - Find all invariant lines under the transformation matrix \(A\).
Fully justify your answer.
12 - Find a line of invariant points under the transformation matrix \(\mathbf { A }\).