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)`

Explanation

To solve the question, we need to analyze the statements related to the invertible matrices \( A \) and \( B \) and identify which one is not correct.

### Step-by-Step Solution:

1. **Understanding the Properties of Invertible Matrices:**
- Recall that a matrix \( A \) is invertible if its determinant \( \text{det}(A) \neq 0 \).
- The inverse of a matrix \( A \) can be expressed as:
\[ A^{-1} = \frac{\text{adj}(A)}{\text{det}(A)} \]
- The adjugate (or adjoint) of a matrix \( A \), denoted as \( \text{adj}(A) \), is a matrix that can be used to find the inverse of \( A \).

2. **Analyzing Option A:**
- The statement is \( \text{adj}(A) = \text{det}(A) \cdot A^{-1} \).
- From the formula for the inverse, we can rearrange it to show:
\[ A^{-1} = \frac{\text{adj}(A)}{\text{det}(A)} \implies \text{adj}(A) = \text{det}(A) \cdot A^{-1} \]
- This statement is correct.

3. **Analyzing Option B:**
- The statement is \( \text{det}(A^{-1}) = \text{det}(A)^{-1} \).
- We know the property of determinants:
\[ \text{det}(A^{-1}) = \frac{1}{\text{det}(A)} \]
- Therefore, this can be rewritten as:
\[ \text{det}(A^{-1}) = \text{det}(A)^{-1} \]
- This statement is also correct.

4. **Analyzing Option C:**
- The statement is \( (A + B)^{-1} = A^{-1} + B^{-1} \).
- The formula for the inverse of a sum of two matrices is not simply the sum of their inverses. The correct formula is:
\[ (A + B)^{-1} \neq A^{-1} + B^{-1} \]
- In fact, the correct expression involves the adjugate and determinant of the sum, and generally, this statement does not hold true.

5. **Conclusion:**
- Since options A and B are correct, and option C is not correct, we conclude that the statement which is not correct is:
\[ (A + B)^{-1} \neq A^{-1} + B^{-1} \]

### Final Answer:
The statement that is not correct is:
**C: \( (A + B)^{-1} = A^{-1} + B^{-1} \)**.