9 The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 4
1 & 5 & - 1
2 & 1 & 5 \end{array} \right)$$ has eigenvalues \(1,5,7\). Find a set of corresponding eigenvectors. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }\).
[0pt] [The evaluation of \(\mathbf { P } ^ { - 1 }\) is not required.]
Determine the set of values of the real constant \(k\) such that \(k ^ { n } \mathbf { A } ^ { n }\) tends to the zero matrix as \(n \rightarrow \infty\).