5 Let \(\mathbf { A } = \left( \begin{array} { l l } 1 & a
0 & 1 \end{array} \right)\), where \(a\) is a positive constant.
- State the type of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
- Prove by mathematical induction that, for all positive integers \(n\),
$$\mathbf { A } ^ { \mathrm { n } } = \left( \begin{array} { c c }
1 & \mathrm { na }
0 & 1
\end{array} \right)$$
Let \(\mathbf { B } = \left( \begin{array} { c c } b & b
a ^ { - 1 } & a ^ { - 1 } \end{array} \right)\), where \(b\) is a positive constant. - Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { n } \mathbf { B }\).