Inverse of product of operators

Question

Let $A$ and $B$ be bounded operators. Suppose that $A$ does not have inverse and that $A$ and $B$ commutes. How can we proof that $AB$ does not has inverse?

Answer 1

If $AB$ be invertable then, let $(AB)^{-1}= C $ so $ABC=I$ that is $BC$ is inverse of $A$ , which is contradiction.