It is given that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\), with corresponding eigenvalue \(\lambda\).
- Write down another eigenvector of \(\mathbf { A }\) corresponding to \(\lambda\).
- Write down an eigenvector and corresponding eigenvalue of \(\mathbf { A } ^ { n }\), where \(n\) is a positive integer.
Let \(\mathbf { A } = \left( \begin{array} { l l l } 3 & 0 & 0
2 & 7 & 0
4 & 8 & 1 \end{array} \right)\). - Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
- Determine the set of values of the real constant \(k\) such that
$$\sum _ { n = 1 } ^ { \infty } k ^ { n } \left( \mathbf { A } ^ { n } - k \mathbf { A } ^ { n + 1 } \right) = k \mathbf { A } .$$
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