Part 1) Let \({A} = \begin{pmatrix} 3 & -2 & 3 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \end{pmatrix},\)
and let \({v}_1 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \quad {v}_2 = \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}, \quad {v}_3 = \begin{pmatrix} 11 \\ 1 \\ -14 \end{pmatrix}.\)
Show that 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.
Part 2) Let n be a positive integer. Use part 1 to find the vectors \(\mathbf{A}^n \mathbf{v}_1, \mathbf{A}^n \mathbf{v}_2, \mathbf{A}^n\mathbf{v}_3.\)
Part 3) Calculate \({A}^{10} \begin{pmatrix} 10 \\ 4 \\ -11 \end{pmatrix}.\)
