JEE Math Practice Question

If $\mathrm{A}, \mathrm{B}, \operatorname{and}\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)$ are non-singular matrices of same order, then the inverse of $A\left(\operatorname{adj}\left(A^{-1}\right)+\operatorname{adj}\left(B^{-1}\right)\right)^{-1} B$, is equal to

1. A.$\frac{A B^{-1}}{|A|}+\frac{B A^{-1}}{|B|}$
2. B.$\operatorname{adj}\left(\mathrm{B}^{-1}\right)+\operatorname{adj}\left(\mathrm{A}^{-1}\right)$
3. C.$\mathrm{AB}^{-1}+\mathrm{A}^{-1} \mathrm{~B}$
4. D.$\frac{1}{|A B|}(\operatorname{adj}(B)+\operatorname{adj}(A))$