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If $\mathrm{A}$ and $\mathrm{B}$ are non-singular matrices and $\operatorname{det}(\mathrm{AB})=(\operatorname{det} \mathrm{A})$ $(\operatorname{det} \mathrm{B})$, then $((\operatorname{det} \mathrm{A})(\operatorname{det} \mathrm{B})) \mathrm{B}^{-1} \mathrm{~A}^{-1}=$
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The correct answer is:
$\operatorname{Adj}(\mathrm{AB})$
$\because \operatorname{det}(A B)=(\operatorname{det} A) \cdot(\operatorname{det} B)$
Now, $(\operatorname{det} A)(\operatorname{det} B)) B^{-1} A^{-1}=(\operatorname{det}(A B))(A B)^{-1}$
$=\operatorname{Adj}(\mathrm{AB}) \quad\left(\because \mathrm{A}^{-1}=\frac{\operatorname{Adj}(\mathrm{A})}{\operatorname{det}(\mathrm{A})}\right)$
Now, $(\operatorname{det} A)(\operatorname{det} B)) B^{-1} A^{-1}=(\operatorname{det}(A B))(A B)^{-1}$
$=\operatorname{Adj}(\mathrm{AB}) \quad\left(\because \mathrm{A}^{-1}=\frac{\operatorname{Adj}(\mathrm{A})}{\operatorname{det}(\mathrm{A})}\right)$
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