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 determine which statement is not correct regarding invertible matrices \( A \) and \( B \), we will analyze each option step by step.

### Step 1: Analyze the first option
**Statement:** Adjoint of \( A \) is equal to \( \text{det}(A) \times A^{-1} \).

**Solution:**
We know that the relationship between the adjoint of a matrix and its inverse is given by:
\[ A^{-1} = \frac{\text{adj}(A)}{\text{det}(A)} \]
Multiplying both sides by \( \text{det}(A) \) gives:
\[ \text{adj}(A) = \text{det}(A) \times A^{-1} \]
Thus, this statement is **correct**.

### Step 2: Analyze the second option
**Statement:** \( \text{det}(A^{-1}) = \text{det}(A)^{-1} \).

**Solution:**
The determinant of the inverse of a matrix is given by:
\[ \text{det}(A^{-1}) = \frac{1}{\text{det}(A)} \]
This can also be expressed as:
\[ \text{det}(A^{-1}) = \text{det}(A)^{-1} \]
Thus, this statement is also **correct**.

### Step 3: Analyze the third option
**Statement:** \( AB^{-1} = B^{-1}A^{-1} \).

**Solution:**
To find \( AB^{-1} \), we can use the property of inverses:
\[ AB^{-1} = A \cdot (B^{-1}) = (AB)^{-1} \]
However, the correct relationship is:
\[ AB^{-1} \neq B^{-1}A^{-1} \]
Instead, we have:
\[ AB^{-1} = A(B^{-1}) = (BA)^{-1} \]
Thus, this statement is **incorrect**.

### Step 4: Analyze the fourth option
**Statement:** \( (A + B)^{-1} = A^{-1} + B^{-1} \).

**Solution:**
This statement is generally not true. The inverse of a sum of matrices is not equal to the sum of their inverses. Thus, this statement is also **incorrect**.

### Conclusion
After analyzing all four options, we find that:
- The first option is correct.
- The second option is correct.
- The third option is incorrect.
- The fourth option is incorrect.

Since the question asks for the statement that is **not correct**, we can conclude that both the third and fourth options are incorrect. However, since the question specifies "which of the following is not correct," we can select the third option as our answer.

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