10 It is given that the eigenvalues of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r } 4 & 1 & - 1
- 4 & - 1 & 4
0 & - 1 & 5 \end{array} \right)$$ are \(1,3,4\). Find a set of corresponding eigenvectors. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { M } ^ { n } = \mathbf { P } \mathbf { D } \mathbf { P } ^ { - 1 }$$ where \(n\) is a positive integer. Find \(\mathbf { P } ^ { - 1 }\) and deduce that $$\lim _ { n \rightarrow \infty } 4 ^ { - n } \mathbf { M } ^ { n } = \left( \begin{array} { r r r } - \frac { 1 } { 3 } & 0 & - \frac { 1 } { 3 }
\frac { 4 } { 3 } & 0 & \frac { 4 } { 3 }
\frac { 4 } { 3 } & 0 & \frac { 4 } { 3 } \end{array} \right)$$