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

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

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