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 $2017$.

  • $\begingroup$ Thank you very much, any any polynomial would work? And could you please explain a little why does this work ? $\endgroup$ May 1 '21 at 8:50
  • $\begingroup$ Yes: if $g(A)=O$ then $g(A)h(A)=O$. $\endgroup$ May 1 '21 at 8:52
  • $\begingroup$ Oh, I totally did not see that. Thank you again :) $\endgroup$ May 1 '21 at 9:00

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