If A and B are square matrics of the same order then which one of the following is always true ?

Options

A. `(A+B)^(-1) =A^(-1) +B^(-1)`
B. adj (AB) = (adj B) (adj A)
C. A and B are non -zero and `|AB|=0 iff |A| =0 ` and |B| =0
D. `(AB)^(-1) =A^(-1) = A^(-1) B^(-1)`

Answer

To determine which statement is always true for square matrices A and B of the same order, we will analyze the options provided in the question.

### Step-by-Step Solution:

1. **Understand the Properties of Matrices**:
- Recall that for square matrices, certain properties hold true, especially regarding addition and multiplication.

2. **Evaluate Option A**:
- The statement is: \( (A + B)^{-1} = A^{-1} + B^{-1} \).
- This is not true in general. The inverse of a sum of matrices is not equal to the sum of their inverses. Therefore, Option A is incorrect.

3. **Evaluate Option B**:
- The statement is: \( (AB)^{-1} = B^{-1}A^{-1} \).
- This is a known property of matrices. The inverse of the product of two matrices is equal to the product of their inverses in reverse order. Therefore, Option B is correct.

4. **Evaluate Option C**:
- The statement is: If A and B are non-zero matrices, then \( \text{det}(A) = \text{det}(B) \).
- This is not necessarily true. Two non-zero matrices can have different determinants. Therefore, Option C is incorrect.

5. **Evaluate Option D**:
- The statement is: \( (AB)^{-1} = A^{-1}B^{-1} \).
- This is incorrect as it should be \( (AB)^{-1} = B^{-1}A^{-1} \). Therefore, Option D is also incorrect.

### Conclusion:
The only statement that is always true among the options given is Option B: \( (AB)^{-1} = B^{-1}A^{-1} \).