Exercise 2
Consider the matrix and vectors
\begin{equation*}
A = \left[\begin{array}{rrr}
5 \amp 2 \amp 2 \\
24 \amp 14 \amp 10 \\
21 \amp 14 \amp 10 \\
\end{array}\right],\qquad
\vvec_1=\threevec{1}{2}{1},
\vvec_2=\threevec{2}{3}{0},
\vvec_3=\threevec{0}{1}{1}
\end{equation*}

Show that the vectors \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\) are eigenvectors of \(A\) and find their associated eigenvalues.
Express the vector \(\xvec = \threevec{0}{3}{4}\) as a linear combination of the eigenvectors.
Use this expression to compute \(A\xvec\text{,}\) \(A^2\xvec\text{,}\) and \(A^{1}\xvec\) as a linear combination of eigenvectors.