Problem

The matrix $A$ is given by

$$A = \begin{pmatrix} 1 & 7 & 11\\ 0 & 2 & 5\\ 0 & 0 & -3 \end{pmatrix}.$$

(a) Find a matrix $P$ and a diagonal matrix $D$ such that $A^6 = PDP^{-1}$.

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

$$A^6 = aA^2 + bA + cI,$$

where $a$, $b$ and $c$ are integers to be determined.