Matrix diagonalizable and eigenvalues

Question

Given the matrix $A∈M_2(\mathbb{Z_7})$

$$ \begin{bmatrix} 1 & 2 \\ 2 & 5 \\ \end{bmatrix} $$

Is it diagonalizable? I think it is because I calculated the eigenvalues which are $4$ and $2$. Since the eigenvalues are two and the order of the matrix is two then the matrix is diagonalizable, right?

Answer 1

Yes, your argument is correct. Perhaps that this will sound pedantic, but you could also have added that, in $\mathbb{Z}_7$, $4\neq2$. That is, you found two distinct eigenvalues,