Let $C$ be a $10\times 10$ matrix whose characteristic polynomial is $(t+2)^5(t-3)^5$. Suppose that $rank((C+2I)^2)=6$ and $rank((C-3I))=8$. What are the possible Jordan forms of $C$?
This is the first problem in which I have ever encountered Jordan forms, so when it says to find all possible Jordan forms of $C$, I take that to mean finding every combination of possible Jordan blocks. Since $rank((C-3I))=8$, I know by rank nullity theorem that the dimension of the eigenspace of $\lambda=3$ is $2$. This tells me that there will be two Jordan blocks for $\lambda=3$, but I don't know how to determine the respective sizes because I don't know what the ranks of higher powers of $(C-3I)$ are. Additionally, I don't know how to glean any information from $rank(C+2I)^2=6$.