Let $\mathsf{F}$ be any field. Let $A$ be an $n \times n$ matrix over $\mathsf{F},$ whose rank is $r \le n.$ Let $\mu \in \mathsf{F}[x]$ be the minimal polynomial of $A.$
What does $\deg(\mu)$ tell about $A$? Is it related to the rank of $A$?
Edit: As noted by Qiaochu, $\deg(\mu)$ is not equal to rank. Are there known cases (or conditions) where $\deg(\mu)$ is actually equal to the rank of $A$?