Proving (AB)^{-1}=B^{-1} A^{-1} in Operator Algebra

[ian2012]

I hope someone can help me with this:

Let the the inverse $$A A^{-1}=A^{-1} A=I$$, where I is the identity operator. Proofing that $$(AB)^{-1}=B^{-1} A^{-1}$$ :

"First, you want to check whether $$(AB)(B^{-1} A^{-1})=I$$. "

However that means the inverse of AB multiplied by AB gives the identity operator, which isn't true, surely, due to Cramer's rule?

 

[Fredrik]

that means the inverse of AB multiplied by AB gives the identity operator, which isn't true

It is. That's just the definition of "inverse".