Commuting Matrics with minimal polynomials, Determine JCF Suppose that $A \in M_{5 \times 5}(\mathbb{C})$ such that $(A - 2I)^{5} = 0$. Suppose that $B \in M_{5 \times 5}(\mathbb{C})$ such that the minimal polynomial of $B$ is $x^{3} + x$ and $AB = BA$. What are the possible Jordan canonical forms of $A$? I know that the minimal polynomial of $A$ divides $(x-2)^{5}$. How can I use the fact that $A,B$ commute to determine restrictions on the minimal polynomial of $A$?

$B$ has three eigenvalues $0$ and $\pm i$.  The generalized eigenspaces of $B$ must be invariant under $A$: therefore $A$ must have at least three Jordan blocks.  These could be
of sizes either $2,2,1$ or $3,1,1$ or $2,1,1,1$ or $1,1,1,1,1$.
