Find ordered pair (α,β)?

Question

Let matrix A = $$ \begin{bmatrix} 1 & 2 & 3 \\ 0 & 5 & 4 \\ 0 & 3 & 2 \\ \end{bmatrix} $$ and $A^3$ - $8A^2$ + αA + βI = O

where I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}

Then the ordered pair (α,β) is:

Answer 1

Compute the characteristic polynomial $p$ of $A$. $p$ has degree $3$ and, by Cayley-Hamilton:

$$p(A)=0.$$