$A+B=AB$ does it follows that $AB=BA$? If $A$, $B$ are two normal operators such that: $A+B=AB$ does it follow that $AB=BA$?

Rearrange the equation to
\begin{eqnarray*}
(A-1)B=A
\end{eqnarray*}
Now assume (A-1) is non singular 
\begin{eqnarray*}
B=(A-1)^{-1}A
\end{eqnarray*}
Geometrically expand $(A-1)^{-1}$ ... so B can be expressed as a sum of powers of A , A will commute with each of these terms, so A commutes with B. 
