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JEE Main 2025 Jan 23 Shift 1, Mathematics Q18: Algebra Of Matrices

JEE Main2025Jan 23, Shift 1Mathematics
Q.

If A, B and are non-singular matrices of same order, then the inverse of is equal to

  1. A

  2. B

  3. C

  4. D

Solution

$ \left[ A\left(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\right)B \right]^{-1} $

$ = B^{-1} \left( \operatorname{adj}(A^{-1}) + \operatorname{adj}(B^{-1}) \right) A^{-1} $

$ = B^{-1}\operatorname{adj}(A^{-1})A^{-1} + B^{-1}\operatorname{adj}(B^{-1})A^{-1} $

$ = B^{-1}|A^{-1}|I + |B^{-1}|IA^{-1} $

$ = \dfrac{B^{-1}}{|A|} + \dfrac{A^{-1}}{|B|} $

$ = \dfrac{\operatorname{adj}B}{|B||A|} + \dfrac{\operatorname{adj}A}{|A||B|} $

$ = \dfrac{1}{|A||B|} \left( \operatorname{adj}B+\operatorname{adj}A \right) $

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