A problem my professor sent out:
Suppose $p$ is a polynomial with constant term nonzero. Suppose $A,B\in M_n(\mathbb{C})$ such that $A=B\cdot p(A)$. Show that $A$ and $B$ commute.
This is a generalization of the problem: suppose $A + B = AB$. Show $A$ and $B$ commute. Here we can note that $(I-A)=(I-B)^{-1}$. I've been trying to adapt that strategy to the more general case, without luck so far.
Note first that if a matrix $M$ is invertible, the use of the characteristic polynomial show that $M^{-1}$ is a polynomial in $M$.
Now we show that $p(A)$ is invertible. Let $x$ such that $p(A)x=0$. Then we get $Ax=Bp(A)x=B0=0$, and then that $ax=0$, where $a$ is the non zero constant term of $p(X)$, because $A^mx=0$ for all $m\geq 1$. Hence $x =0$, and $p(A)$ is invertible. Now apply the remark with $M=p(A)$: $M^{-1}$ is a polynomial in $M$, hence in $A$, and $B=AM ^{-1}$ is also a polynomial in $A$. Hence $B$ commute to $A$.