I'm a TA of the linear algebra course for undergraduate freshmen. There is an exercise in the textbook:
Let $A$ be a complex matrix of size $n\times n$. For every polynomials (remark: we only consider polynomials over $\mathbb{C}$) $p(x)$ and $q(x)$, let $f(x)=p(x)+q(x),g(x)=p(x)q(x)$. Prove that $f(A)=p(A)+q(A),g(A)=p(A)q(A)$.
What does it mean? As abstract functions from $M_n(\mathbb{C})$ to $M_n(\mathbb{C})$, these consequences should be automatically satisfied. I don't see there is anything to "proof".