Suppose a matrix $A \in \text{Mat}_{2\times 2}(\mathbb{F}_5)$ has characteristic polynomial $x^2 - x +1$. Is $A$ diagonalizable over $\mathbb{F}_5$?

Question

Suppose a matrix $A \in \text{Mat}_{2\times 2}(\mathbb{F}_5)$ has characteristic polynomial $x^2 - x +1$. Is $A$ diagonalizable over $\mathbb{F}_5$?

Normally, I would just check to see if the geometric multiplicity and algebraic multiplicity are equal for each eigenspace, but over $\mathbb{F}_5$, I am not even sure what the eigenvalues are!

Answer 1

The char polynomial has no roots in $\Bbb{F}_5$, so the matrix is not diagonalizable over that field. It is diagonalizable over $\Bbb{F}_{25}$ because the char polynomial has two different roots in that field.

Answer 2

If your matrix were diagonalizable, then there would be a diagonal matrix $$\Lambda = \begin{bmatrix}\lambda_1 & 0 \\ 0 & \lambda_2\end{bmatrix} \in \text{Mat}_{2 \times 2}(\mathbb F_5)$$ with the same characteristic polynomial. But then, $\Lambda^2 - \Lambda + I = 0_{2\times 2}$ implies $\lambda_i^2 - \lambda_i + 1 = 0$ for $i=1,2$, and there are no such elements in $\mathbb F_5$.