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KCET2026MathematicsMatrices

If A and B are invertible matrices of same order, then which of the following is not correct?

Options

  1. AA ( adj A) = ( adj A) A = A I
  2. BA ( adj A) = ( adj A) A = |A |I
  3. C(AB)⁻¹ = B⁻¹ A⁻¹
  4. D|A| 0, |B| 0

Correct answer

A. A ( adj A) = ( adj A) A = A I

Step-by-step solution

For any square matrix A of order n , the fundamental property relating the matrix and its adjoint is A ( adj A) = ( adj A) A = |A| I , where I is the identity matrix of order n . Thus, the statement A ( adj A) = ( adj A) A = A I is incorrect. Since A and B are invertible matrices, their determinants must be non-zero, which means |A| 0 and |B| 0 . The reversal law for the inverse of the product of two invertible matrices states that (AB)⁻¹ = B⁻¹ A⁻¹ . Therefore, the only incorrect statement is the first one. Answer:

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