i have a question and i got stuck on certain step of the solution.

i'll write the question and the solution till the point where i got stuck

and explain why i got stuck

the question:

there is A 3x3 matrices which have 3 real eigenvalues,

so the jordan form of

find the caracteristic polinomial

and the minimal polinomial of A and write its jordan form.

the solution:

it is given that is similar to .

so the caracteristic polinomial of is

so 8 and 1 are the only eigenvalues.

suppose and so

if and its eigenvalue of A then is eigen value of

so

so

so the possible caracteristic polinomials are:

suppose that or

then the caracteristic polinomial will be or

and thats cannot happen .(stuck on understanding this point)

my probem with this point:

if i know that 8 is eigenvalue of but why the

why we have the same structure??

another problem is why these possible caracteristic polinomials cannot work?