Exercise 1

Consider the matrix and vectors $$A = \left[\begin{array}{rr} 8 & -10 \\ 5 & -7 \\ \end{array}\right],\qquad \vvec_1=\twovec{2}{1},~~~ \vvec_2=\twovec{1}{1}$$.

1. Show that $\vvec_1$ and $\vvec_2$ are eigenvectors of $A$ and find their associated eigenvalues.

2. Express the vector $\xvec = \twovec{-4}{-1}$ as a linear combination of $\vvec_1$ and $\vvec_2$.

3. Use this expression to compute $A\xvec$, $A^2\xvec$, and $A^{-1}\xvec$ as a linear combination of eigenvectors.