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).

Answer 2

We are given that A and B are invertible matrices and we need to identify which of the following statements is not correct.

1. adj A = |A| A^-1 This is a correct statement. We know that \(A (\text{adj } A) = |A|I\), where \(I\) is the identity matrix. Multiplying both sides by \(A^{-1}\), we get \((\text{adj } A) = |A| A^{-1}\).
2. det (A^-1) = [det (A)]^-1 This is also a correct statement. We know that \(A A^{-1} = I\), so \(|A A^{-1}| = |I| = 1\). Thus, \(|A| |A^{-1}| = 1\), which implies \(|A^{-1}| = \frac{1}{|A|} = [det(A)]^{-1}\).
3. (AB)^-1 = B^-1A^-1 This is also a correct statement. We have \((AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = AIA^{-1} = AA^{-1} = I\). Similarly, \((B^{-1}A^{-1})(AB) = B^{-1}(A^{-1}A)B = B^{-1}IB = B^{-1}B = I\). Therefore, \((AB)^{-1} = B^{-1}A^{-1}\).
4. (A + B)^-1 = B^-1+ A^-1 This is not a correct statement in general. The inverse of a sum is not generally the sum of the inverses. There is no general formula for the inverse of a sum of matrices.

Therefore, the incorrect statement is \((A + B)^{-1} = B^{-1} + A^{-1}\).