How to calculate the diagonal matrix The question says: find the eigenvalues and corresponding eigenvectors of the matrix $A$. This I could do. But then it says: hence find a non-singular matrix $P$ and a diagonal matrix $D$ such that $A + A^2 + A^3 = PDP^{-1}$ , where 
$$
        A =\begin{pmatrix}
        6 & 4 & 1 \\
        -6 & -1 & 3 \\
        8 & 8 & 4 \\
        \end{pmatrix}
$$
The eigenvalues are $-1, 2, 8$ with corresponding eigenvectors $\begin{pmatrix}
        4 \\
        -9 \\
        8 \\
        \end{pmatrix}$,
        $\begin{pmatrix}
        5 \\
        -6 \\
        4 \\
        \end{pmatrix}$,
$\begin{pmatrix}
        1 \\
        0 \\
        2 \\
        \end{pmatrix}$
I cannot calculate $A + A^2 + A^3$ numerically, so how can I do this?

You can find $Q$ such that $A=QDQ^{-1}$ with $D$ diagonal, right? Then show that $$A+A^2+A^3=Q(D+D^2+D^3)Q^{-1}$$
