Chapter 5.6, Problem 1PP

The matrix A below has eigenvalues 1, $\frac{2}{3}$, and $\frac{1}{3}$, with corresponding eigenvectors v₁, v₂. and v₃:

A = $\frac{1}{9}\left[\begin{array}{ccc}7& -2& 0\\ -2& 6& 2\\ 0& 2& 5\end{array}\right]$, v₁ = $\left[\begin{array}{l}-2\\ 2\\ 1\end{array}\right]$, v₂ = $\left[\begin{array}{l}2\\ 1\\ 2\end{array}\right]$, v₃ = $\left[\begin{array}{l}1\\ 2\\ -2\end{array}\right]$