Suppose $A,B$ are $3\times 3$ matrices with minimal polynomials $x^2-4$ and $x+2$ resp. What are the possible dimensions of the kernel of $A-B$?
The possible JCFs for $A$ are $(2,2,-2), (2,-2,-2)$. The JCF of $B$ is $(-2,-2,-2)$. Thus the possible JCFs for $A-B$ are $(4,4,0),(4,0,0)$, and the possible dimensions are $1$ or $2$. (They are invariant under conjugation.)
Is this a correct sketch of solution?