If $\mathrm{A}$ and $\mathrm{B}$ are non-singular matrices and…

If $\mathrm{A}$ and $\mathrm{B}$ are non-singular matrices and $\operatorname{det}(\mathrm{AB})=(\operatorname{det} \mathrm{A})$ $(\operatorname{det} \mathrm{B})$, then $((\operatorname{det} \mathrm{A})(\operatorname{det} \mathrm{B})) \mathrm{B}^{-1} \mathrm{~A}^{-1}=$
  1. Adj (BA)
  2. $\operatorname{Adj}(\mathrm{A})+\operatorname{Adj}(\mathrm{B})$
  3. $\operatorname{Adj}(\mathrm{AB})$
  4. $(\operatorname{Adj} B)(\operatorname{Adj} A)$

Solution

$\because \operatorname{det}(A B)=(\operatorname{det} A) \cdot(\operatorname{det} B)$ Now, $(\operatorname{det} A)(\operatorname{det} B)) B^{-1} A^{-1}=(\operatorname{det}(A B))(A B)^{-1}$ $=\operatorname{Adj}(\mathrm{AB}) \quad\left(\because \mathrm{A}^{-1}=\frac{\operatorname{Adj}(\mathrm{A})}{\operatorname{det}(\mathrm{A})}\right)$

Asked in: AP EAMCET 2023 (18 May Shift 1)

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