Problem

The matrix $A$ is given by

$$A = \begin{pmatrix} -2 & 0 & 0\\ 0 & 7 & 9\\ 4 & 1 & 7 \end{pmatrix}.$$

(a) Show that the characteristic equation of $A$ is

$$\lambda^3 - 12\lambda^2 + 12\lambda + 80 = 0$$

and find the eigenvalues of $A$.

(b) Use the characteristic equation of $A$ to show that

$$A^4 = pA^2 + qA + rI,$$

where $p$, $q$ and $r$ are integers to be determined.

(c) Find a matrix $P$ and a diagonal matrix $D$ such that $(A - 3I)^4 = PDP^{-1}$.