a 2×2 matrix that has a power equal to the identity and an eigenvalue in F_p, but is not diagonalizable?

Question

The problem states: Find a 2×2 matrix with entries in $\mathbb{F}_p$ that has a power equal to the identity and an eigenvalue in $\mathbb{F}_p$, but is not diagonalizable. I have found some examples with specific p, but don't know how to generalize them to any prime number p.

Answer 1

Hint. Have a look at $$ A := \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} =: \mathrm{Id} + N $$ $A$ is surely not diagonalizable, but has $1$ as an eigenvalue. Now look at powers of $A$ (the power that is the identity will depend on $p$).