Exercise 1.6.2

The eigenvalues of $A$ are: ${\lambda }_1=2$, ${\lambda }_2=-1$. The eigenvectors are $v_1=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$, and $v_2=\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 1\hfill \end{array}\right]$.

The eigenvalues of $A^{-1}$ are: ${\lambda }_1=\frac{1}{2}$, ${\lambda }_2=-1$. The eigenvectors are $v_1=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$, and $v_2=\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 1\hfill \end{array}\right]$

$A^{-1}$ has the same eigenvectors as $A$. When $A$ has eigenvalues ${\lambda }_1$ and ${\lambda }_2$, its inverse has eigenvalues ${\lambda }_1^{-1}$, and ${\lambda }_2^{-1}$.