Exercise 2

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

1. Show that the vectors $\vvec_1$, $\vvec_2$, and $\vvec_3$ are eigenvectors of $A$ and find their associated eigenvalues.

2. Express the vector $\xvec = \threevec{0}{-3}{-4}$ as a linear combination of the eigenvectors.

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