Question

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

• A. adjA = |A| A^-1
• B. det (A^-1) = [det (A)]^-1
• C. (AB)^-1 = B^-1A^-1
• D. (A + B)^-1 = B^-1+ A^-1

Answer 1

The Correct Option is D

We are given that \( A \) and \( B \) are invertible matrices.
We need to determine which of the following statements is not correct:

(A) \( \text{adj}(A) = |A| A^{-1} \)
This is a correct property of adjoint matrices. It holds true that the adjoint of \( A \) is equal to \( |A| \) multiplied by the inverse of \( A \).

(B) \( \det(A^{-1}) = [\det(A)]^{-1} \)
This is also a correct property of determinants. The determinant of the inverse of \( A \) is indeed the reciprocal of the determinant of \( A \).

(C) \( (AB)^{-1} = B^{-1} A^{-1} \)
This is a correct property of the inverse of the product of matrices. The inverse of the product \( AB \) is equal to the product of the inverses \( B^{-1} A^{-1} \), but in reverse order.

(D) \( (A + B)^{-1} = B^{-1} + A^{-1} \)
This statement is incorrect. The inverse of the sum of two matrices \( (A + B)^{-1} \) is not equal to the sum of the individual inverses \( B^{-1} + A^{-1} \). This is a common misconception, and it does not hold for matrix operations.

The correct answer is (D).