(a) Let \[\bold{A} = \begin{pmatrix} 3 & -2 & 3 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \end{pmatrix},\]and let
\[\bold{v}_1 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \quad \bold{v}_2 = \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}, \quad \bold{v}_3 = \begin{pmatrix} 11 \\ 1 \\ -14 \end{pmatrix}.\]Show that $\mathbf{A}$ sends each of $\mathbf{v}_1, \mathbf{v}_2,$ and $\mathbf{v}_3$ to scalar multiples of themselves, and find the value of these scalars.

(b) Let $n$ be a positive integer. Use part (a) to find the vectors \[\mathbf{A}^n \mathbf{v}_1, \mathbf{A}^n \mathbf{v}_2, \mathbf{A}^n\mathbf{v}_3.\](c) Write the vector $\begin{pmatrix} 10 \\ 4 \\ -11 \end{pmatrix}$ as a linear combination of $\mathbf{v}_1, \mathbf{v}_2,$ and $\mathbf{v}_3$.

(d) Using parts (a), (b), and (c), calculate
\[\bold{A}^{10} \begin{pmatrix} 10 \\ 4 \\ -11 \end{pmatrix}.\]