It is given that
\(\mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & 1 \\ 0 & -2 & 1 \\ 0 & 0 & 1 \end{array}\right) .\)
(i) Find the eigenvalue of \(\mathbf{A}\) corresponding to the eigenvector \(\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)\).

(ii) Write down the negative eigenvalue of \(\mathbf{A}\) and find a corresponding eigenvector.

(iii) Find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf{A}+\mathbf{A}^{6}\).