3
$\begingroup$

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

$\endgroup$
1
7
$\begingroup$

All you need to do is multiply $(X-5)(X+1)^2$ by any polynomial of degree $2017$.

$\endgroup$
3
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.