I'm currently studying Cayley-Hamilton proof, but I think I'm missing something massive.
The theorem states that an endomorphism $\phi$ cancels its characteristic polynomial. I don't understand why the proof is not trivial.
Let $A$ be a matrix of $\phi$ in a basis $\mathcal{B}$. We know that the characteristic polynomial of $A$ is the same as $\phi$ no matter which basis of the vector space we take. Then, $$P_A(A)=\det(A-AI)=\det(A-A)=0$$
So the proof should be trivial, I don't understand where my error is.