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

Options

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

Answer

To determine which statement about invertible matrices A and B is not correct, we will analyze each option systematically.

### Step-by-Step Solution:

1. **Understanding Invertible Matrices**:
- A matrix is said to be invertible if there exists another matrix such that their product is the identity matrix. If A is invertible, it has an inverse denoted as \( A^{-1} \), and similarly for B.

2. **Option Analysis**:
- **Option 1**: \( A = A^{-1} \) exists.
- This statement is true because if A is invertible, then its inverse \( A^{-1} \) exists. Therefore, this option is correct.

- **Option 2**: \( \text{det}(A^{-1}) = \frac{1}{\text{det}(A)} \).
- This statement is also true. The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix itself. Hence, this option is correct.

- **Option 3**: \( (AB)^{-1} = B^{-1}A^{-1} \).
- This statement is true as well. The inverse of the product of two matrices is the product of their inverses in reverse order. Thus, this option is correct.

- **Option 4**: \( (A + B)^{-1} = A^{-1} + B^{-1} \).
- This statement is false. The inverse of a sum of two matrices is not equal to the sum of their inverses. The correct expression for the inverse of a sum is more complex and does not simplify to this form. Therefore, this option is not correct.

3. **Conclusion**:
- The statement that is not correct is **Option 4**: \( (A + B)^{-1} \neq A^{-1} + B^{-1} \).

### Final Answer:
The option that is not correct is **Option 4**: \( (A + B)^{-1} \neq A^{-1} + B^{-1} \).