The matrix
\(\mathbf{A}=\left(\begin{array}{rrr} 3 & 1 & 4 \\ 1 & 5 & -1 \\ 2 & 1 & 5 \end{array}\right)\)
has eigenvalues \(1,5,7\). Find a set of corresponding eigenvectors.

Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{n}=\mathbf{P D P}^{-1}\).
[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 \(n \rightarrow \infty\).