Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2015 / 2 / a / iv

In the finite field ${\mathbb{F}}_2$, the matrix becomesOn the first two coordinate vectors it is a nonzero nilpotent Jordan block of size two, and on the last it is the identity map. ConsequentlyThe repeated zero root prevents diagonalizability. Directly, the only possible eigenvalues are zero and one, so diagonalizability would imply $A^2=A$, whereas the displayed off-diagonal entry disappears on squaring.