Problem

The matrix $A$ is given by

$$A=\begin{pmatrix} -11&1&8\\ 0&-2&0\\ -16&1&13 \end{pmatrix}.$$

(a) Show that $\begin{pmatrix}1\\1\\1\end{pmatrix}$ is an eigenvector of $A$ and state the corresponding eigenvalue.

(b) Show that the characteristic equation of $A$ is $\lambda^3-19\lambda-30=0$ and hence find the other eigenvalues of $A$.

(c) Use the characteristic equation of $A$ to find $A^{-1}$.