Help with commutative property of matrices problem

Given matrices $A$ and $B$, where that $AB = A + B$, prove $AB = BA$.

I keep coming up with AB = AB. It seems like basic algebra, but for the life of me, I'm getting nowhere :/. Someone help please?

Solution 1:

Hint: Consider $(A-I)(B-I)$, where $I$ is the identity matrix.

Solution 2:

Assume for now that 1 is not an eigenvalue of $B$. Then $B-I$ is invertible, so from the assumption we get $A=B (B-I)^{-1}$. $(B-I)^{-1}$ commutes with $B$ since it commutes with $B-I$ and with $I$.

EDIT: Now I claim that 1 can't be an eigenvalue of $B$. Indeed, suppose $Bv=v$ for some vector $v$. Then $ABv=Av+Bv$, hence $Av=Av+v$, hence $v=0$.