# Finding polynomial to the power of 2020

I am trying to solve a homework problem, but I am stuck at a point where I don't know what am I suppose to do next.

We're given a $$3 \times 3$$ matrix

$$A = \begin{pmatrix}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{pmatrix}$$

And we have to find a polynomial $$p(x)$$ such that deg($$p(x)$$) = 2020, and $$p(A) = 0$$ (as 0 matrix 3x3)

What I did was finding the characteristic polynomial which is $$p(x) = (x-5)(x+1)^2$$

And I know that if I use Cayley Hamilton I can place A in the characteristic polynomial and get the zero matrix. but what is that part with the degree 2020, I don't understand how do I do that or what do I rely on?

This might seem easy but I really can't see it. any help is appreciated :)
Thank you

All you need to do is multiply $$(X-5)(X+1)^2$$ by any polynomial of degree $$2017$$.
• Yes: if $g(A)=O$ then $g(A)h(A)=O$. May 1 '21 at 8:52