Hi, I am not 100% sure my working is correct for this question. Could someone be kind enough to check my working please.

$\displaystyle F= \{a+bi | a,b \in F_3 \}$

$\displaystyle A=\dbinom{2 \ \ 1}{1 \ \ 1} \in F^{2x2}$

The characteristic polynomial of A:

$\displaystyle det \dbinom{2-\lambda \ \ \ \ 1}{1 \ \ \ \ \ 1-\lambda}= \lambda -3\lambda +1$

So the eigenvalues are $\displaystyle \lambda_1 = \frac{-(\sqrt{5}-3)}{2}$ and $\displaystyle \lambda_2 = \frac{\sqrt{5}+3}{2}$.

To find the eigenvectors we sub in $\displaystyle \lambda_1$ and $\displaystyle \lambda_2$ into the equations:

$\displaystyle (2-\lambda)x+y=0$

$\displaystyle x+(1-\lambda)y=0$

to solve for x and y for both eigenvalues.

So for example, for $\displaystyle \lambda_2$ I get $\displaystyle y=\frac{3(\sqrt{5}+2)}{2}$ and $\displaystyle x=\frac{3(6\sqrt{5}+13)}{4}$.

Am I correct in my working out so far?

Thanks