Let $A,B\in M_2(\mathbb R)$. Prove that there's $C \in M_2(\mathbb R)$ that is not sum of $f(A)+g(B)$ for any polynomials $f,g \in \mathbb R[X]$.
I know that if $\lambda $ is eigenvalue of $A$ then $f(\lambda)$ of $f(A)$ and also for $B$ and $g$.
What else should I do?
Thank you.