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Let $A$ be a $3 \times 3$ matrix with eigenvalues $\lambda_{1}, \lambda_{2}, \lambda_{3}$ and corresponding linearly independent eigenvectors $V_{1}, V_{2}, V_{3}$ which we can therefore use as a basis.

a) If $X=a V_{1}+b V_{2}+c V_{3}$, compute $A X, A^{2} X$, and $A^{35} X$ in terms of $\lambda_{1}, \lambda_{2}, \lambda_{3}$, $V_{1}, V_{2}, V_{3}, a, b$ and $c$ (only).
b) If $\lambda_{1}=1,\left|\lambda_{2}\right|<1$, and $\left|\lambda_{3}\right|<1$, compute $\lim _{k \rightarrow \infty} A^{k} X$. Explain your reasoning clearly.
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