Let \(\mathbf{A} = \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix}\), where \(a\) is a positive constant.

(a) State the type of the geometrical transformation in the \(x-y\) plane represented by \(\mathbf{A}\). [1]

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

\(\mathbf{A}^n = \begin{pmatrix} 1 & na \\ 0 & 1 \end{pmatrix}.\) [5]

Let \(\mathbf{B} = \begin{pmatrix} b & b \\ a^{-1} & a^{-1} \end{pmatrix}\), where \(b\) is a positive constant.

(c) Find the equations of the invariant lines, through the origin, of the transformation in the \(x-y\) plane represented by \(\mathbf{A}^n \mathbf{B}\). [6]