Chapter 5.6, Problem 1E

Let A be a 2 × 2 matrix with eigenvalues 3 and 1/3 and corresponding eigenvectors v₁ = $\left[\begin{array}{l}1\\ 1\end{array}\right]$ and v₂ = $\left[\begin{array}{l}-1\\ 1\end{array}\right]$. Let {x_k} be a solution of the difference equation x_k+1 = Ax_k, x₀ = $\left[\begin{array}{l}9\\ 1\end{array}\right]$.

1. a. Compute x₁ = Ax₀. [Hint: You do not need to know A itself.]
2. b. Find a formula for x_k involving k and the eigenvectors v₁ and v₂.