JEE Main 2025 · 23 January · morning · Single correct · +4 / −1

If A, B, and(adj(A⁻¹)+adj(B⁻¹)) are non-singular matrices of same order, then the inverse of A(adj(A⁻¹)+adj(B⁻¹))⁻¹ B, is equal to

  1. (a) (A B⁻¹)/|A|+(B A⁻¹)/|B|
  2. (b) adj(B⁻¹)+adj(A⁻¹)
  3. (c) AB⁻¹+A⁻¹ B
  4. (d) 1/(|A B|)(adj(B)+adj(A))
Answer and solution

Answer (d) 1/(|A B|)(adj(B)+adj(A))

Solution

Maths sir · 4 steps

Undo the product in reverse order, then turn each adj of an inverse back into the matrix over its determinant.

  1. Reverse the order. Call M = adj(A⁻¹) + adj(B⁻¹). Since (PQR)⁻¹ = R⁻¹Q⁻¹P⁻¹, the inverse of AM⁻¹B is B⁻¹MA⁻¹.
  2. Adj of an inverse. For any invertible X, adj(X) = |X| X⁻¹. Put X = A⁻¹: adj(A⁻¹) = |A⁻¹| A = A/|A|. In the same way, adj(B⁻¹) = B/|B|.

    B⁻¹(A/|A|+B/|B|)A⁻¹ = B⁻¹/|A|+A⁻¹/|B|

  3. Bring in adjoints. Write B⁻¹ = adj(B)/|B| and A⁻¹ = adj(A)/|A|. Both terms now have |A||B| = |AB| in the denominator.
  4. So. The inverse is 1/|AB|(adj(B)+adj(A)), which is option (d).

The idea to keep. Remember adj(A⁻¹) = A/|A| and A⁻¹ = adjA/|A|. When inverting a product, reverse the order first, then let the middle factors cancel.

Answer from the NTA final answer key. Solution by PrepWiser.

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Asked about 110 times in JEE Main since 2020. This question is from 2025.