To find the Jordan Canonical Form

Question

Consider a matrix A (5x5) with all entries = 1. Here the entries are considered as elements of $F_5$ ,the finite field of order 5.

What is the Jordan canonical form?

I have found out that $A^2=0$ and thus the minimal polynomial is $x^2$. So I know there are (two 2x2 blocks and one 1x1 block) OR (one 2x2 block and three 1x1 blocks)

How do I tell which?

Answer 1

Hint. Consider the rank of $A$.

Answer 2

Hint: If $A$ is an $n \times n$ matrix, then $n - \operatorname{rk}(A)$ is the total number of Jordan blocks that $A$ has associated with $\lambda = 0$.