It is given that $\mathbf { e }$ is an eigenvector of the matrix $\mathbf { A }$, with corresponding eigenvalue $\lambda$.\\
(i) Write down another eigenvector of $\mathbf { A }$ corresponding to $\lambda$.\\

(ii) Write down an eigenvector and corresponding eigenvalue of $\mathbf { A } ^ { n }$, where $n$ is a positive integer.\\

Let $\mathbf { A } = \left( \begin{array} { l l l } 3 & 0 & 0 \\ 2 & 7 & 0 \\ 4 & 8 & 1 \end{array} \right)$.\\
(iii) Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$.\\

(iv) Determine the set of values of the real constant $k$ such that

$$\sum _ { n = 1 } ^ { \infty } k ^ { n } \left( \mathbf { A } ^ { n } - k \mathbf { A } ^ { n + 1 } \right) = k \mathbf { A } .$$

If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\

\hfill \mbox{\textit{CAIE FP1 2018 Q11 OR}}