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 $A^{-1}$.