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Prove that the inverse of a product is the product of the inverses

Prove that (ab)^{-1} = a^{-1} b^{-1} if a, b \neq 0.


Proof. Since a, b \neq 0 we know there exist elements a^{-1}, b^{-1} \in \mathbb{R} such that aa^{-1} = 1 and bb^{-1} = 1. So,

    \begin{align*}    (ab)^{-1} &= (ab)^{-1} \cdot (aa^{-1}) \cdot (bb^{-1}) \\    &= (ab)^{-1} \cdot (ab) \cdot (a^{-1} b^{-1}) & (\text{Using assoc. and comm.}) \\    &= a^{-1} b^{-1}. \qquad \blacksquare \end{align*}

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