If A and B are invertible matrices of same order, then which of the following statement is not true ?

Options

A. A) `|A^(-1)|=|A|^(-1)`
B. `B) "adj "A=|A|A^(-1)`
C. `(A+B)^(-1)=B^(-1)+A^(-1)`
D. D) (AB)-1=B^(-1)A^()-1

Answer

To solve the problem, we need to analyze the statements regarding the invertible matrices \( A \) and \( B \) of the same order and identify which statement is not true.

### Step-by-Step Solution:

1. **Understanding Invertible Matrices**:
- A matrix \( A \) is said to be invertible if there exists a matrix \( A^{-1} \) such that \( A A^{-1} = I \), where \( I \) is the identity matrix.

2. **Analyzing the Statements**:
- **Option 1**: \( A^{-1} = \text{det}(A)^{-1} \cdot \text{adj}(A) \)
- This statement is true. The formula for the inverse of a matrix \( A \) is given by \( A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \), where \( \text{adj}(A) \) is the adjugate of \( A \).

- **Option 2**: \( A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \)
- This statement is also true as it directly follows from the properties of determinants and adjugates.

- **Option 3**: \( (A + B)^{-1} = A^{-1} + B^{-1} \)
- This statement is **not true**. The inverse of a sum of matrices is not equal to the sum of their inverses. The correct relation is \( (A + B)^{-1} \neq A^{-1} + B^{-1} \).

- **Option 4**: \( (AB)^{-1} = B^{-1}A^{-1} \)
- This statement is true. The inverse of the product of two matrices is the product of their inverses in reverse order.

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

### Final Answer:
The statement that is not true is **Option 3**: \( (A + B)^{-1} \neq A^{-1} + B^{-1} \).