Exercise 1
Consider the matrix and vectors
\begin{equation*}
A = \left[\begin{array}{rr}
8 \amp 10 \\
5 \amp 7 \\
\end{array}\right],\qquad
\vvec_1=\twovec{2}{1},
\vvec_2=\twovec{1}{1}\text{.}
\end{equation*}

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