Show that a matrix is not diagonalizable over a finite field

Question

I have a matrix: $$\ \left[ {\begin{array}{cc} 2 & 2 \\ 1 & 2 \\ \end{array} } \right] $$ which I need to show that it cannot be diagonalized over the finite field $\mathbb{F_3}$.

I have computed the characteristic polynomial. It is: $$p(x)=x^2-4x+2.$$

Obviously in $\mathbb{F_3}$, $4=1.$ Hence, $$p(x)=x^2-x+2.$$ But $p(x)$ does not split into linear factors in $\mathbb{F_3}$. Is this the proof?

Answer 1

Hint: A few basics about reduction: if a matrix is diagonalizable, it has eigenvalues. There are necessarily roots of the characteristic polynomial.