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Mathematics · Matrices

JEE Main 2025 — 23 January, Morning Shift — Question 18

If A,B\mathrm{A}, \mathrm{B} and (adj⁡(A−1)+adj⁡(B−1))\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(adj⁡(A−1)+adj⁡(B−1))−1 B\mathrm{A}\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)^{-1} \mathrm{~B}, is equal to

  1. Option A:

    AB−1+A−1 B\mathrm{AB}^{-1}+\mathrm{A}^{-1} \mathrm{~B}

  2. Option B:

    adj⁡(B−1)+adj⁡(A−1)\operatorname{adj}\left(\mathrm{B}^{-1}\right)+\operatorname{adj}\left(\mathrm{A}^{-1}\right)

  3. Option C:

    1∣AB∣(adj⁡(B)+adj⁡(A))\frac{1}{|\mathrm{AB}|}(\operatorname{adj}(\mathrm{B})+\operatorname{adj}(\mathrm{A}))

    Correct
  4. Option D:

    AB−1∣A∣+BA−1∣B∣∣B∣\frac{A B^{-1}}{|A|}+\frac{{B A^{-1}}^{|B|}}{|B|}

Answer: C

Step-by-step solution

[A(adj⁡(A−1)+adj⁡(B−1))−1⋅ B]−1\left[\mathrm{A}\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)^{-1} \cdot \mathrm{~B}\right]^{-1}

B−1⋅(adj⁡( A−1)+adj⁡(B−1))⋅A−1\mathrm{B}^{-1} \cdot\left(\operatorname{adj}\left(\mathrm{~A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right) \cdot \mathrm{A}^{-1}

B−1adj⁡( A−1)A−1+B−1(adj⁡( B−1))⋅A−1\mathrm{B}^{-1} \operatorname{adj}\left(\mathrm{~A}^{-1}\right) \mathrm{A}^{-1}+\mathrm{B}^{-1}\left(\operatorname{adj}\left(\mathrm{~B}^{-1}\right)\right) \cdot \mathrm{A}^{-1}

B−1∣ A−1∣I+∣B−1∣IA−1\mathrm{B}^{-1}\left|\mathrm{~A}^{-1}\right| \mathrm{I}+\left|\mathrm{B}^{-1}\right| \mathrm{IA}^{-1}

B−1∣ A∣+A−1∣ B∣\frac{\mathrm{B}^{-1}}{|\mathrm{~A}|}+\frac{\mathrm{A}^{-1}}{|\mathrm{~B}|}

⇒adjB⁡∣B∣∣A∣+adj⁡A∣A∣∣B∣\Rightarrow \frac{\operatorname{adjB}}{|\mathrm{B}||\mathrm{A}|}+\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}||\mathrm{B}|}

=1∣ A∣∣B∣(adjB⁡+adj⁡A)=\frac{1}{|\mathrm{~A}||\mathrm{B}|}(\operatorname{adjB}+\operatorname{adj} \mathrm{A})

Answer key and solution verified before publishing.

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Exam
JEE Main 2025
Paper
23 January, Morning Shift
Subject
Mathematics
Chapter
Matrices
Topic
Inverse of a Matrix

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If A , B and ( adj ( A -1 )+ adj ( B -1 ) ) are non-singular matrices… | JEE Main 2025 PYQ with Solution · DhiX AI