LA2.2.2

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Let C=AB then C is invertible: A and B are invertibles so \det {A}\neq 0 and \det {B}\neq 0 therefore \det {C}=\det {A}\det {B}\neq 0 and so C^{{-1}} exists.
On the other side, C=AB\Rightarrow A^{{-1}}C=B\Rightarrow A^{{-1}}=BC^{{-1}}\Rightarrow B^{{-1}}A^{{-1}}=C^{{-1}}. QED.