set G equal to what?

Question

My attempts : i construct a matrix $A= \begin{bmatrix} 1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&2&1&0&0&0&0\\ 0&0&0&2&0&0&0&0\\ 0&0&0&0&2&0&0&0\\ 0&0&0&0&0&2&0&0\\ 0&0&0&0&0&0&2&0\\ 0&0&0&0&0&0&0&2\end{bmatrix}$

Now we know that the characteristic polynomial of A is $(x − 1)^2 (x − 2)^6$ and the minimal polynomial of A is$ (x − 1)(x − 2)^2$

Now we will take only eigenvalue $ 2$ of $ A$ that is characteristic of eigenvalue $2 =(x-2)^6$ and the minimial of eigenvalue $2 = (x-1)^2$

Now set G will be equal to $ 6- 2 = 4$

so $G = 4$

Is my answer is correct or nots? pliz reply

i have my exam tommorrows. so i need its urgently any Hints/solution

Thanks in advance i would be more thankful

Answer 1

In the matrix $A$ we have an eigenvalue $2$ which has geometric multiplicity $4$ and one which has geometric multiplicity $1$ (the one of the jordan block $2-by-2$). Thus we have that $G=\{1,4\}$.