Suppose $S,T:\mathbb C^3\to\mathbb C^3$ are linear operators. The degree of the minimal polynomial of each of the operators is at most 2. Show that they share a common eigenvector.
I tried to exploit the condition on the degree. I obtained that there are two possible Jordan canonical forms for $S,T$. The first possibility is that the JCF is diagonal of the form $(a,a,b)$. The second possibility is that the JCF has one block of size 2 with eigenvalue $a$ and 1 block of size 1 with eigenvalue $a$. For the other operator the possibilities for the JCF are the same except that the eigenvalues may be different. But I don't know how to proceed from this point.