If A and B are square matrices of same order, then

Options

A. `A-B=B-A`
B. A+B=B-A
C. `A+B=B+A`
D. `AB=BA`

Answer

To solve the question, we need to analyze the properties of square matrices A and B of the same order. We will evaluate the given statements one by one.

### Step-by-Step Solution:

1. **Understanding Matrix Subtraction**:
- For any two matrices A and B of the same order, the subtraction is defined as:
\[ A - B = A + (-B) \]
- Here, \(-B\) is the additive inverse of matrix B.

2. **Evaluating the First Option**:
- The first option states that \( A - B = B - A \).
- Rearranging this gives us:
\[ A - B + B = B - A + B \implies A = 2B - A \]
- This is not generally true. Therefore, the first option is **incorrect**.

**Hint**: Remember that subtraction of matrices is not commutative, meaning \( A - B \neq B - A \).

3. **Evaluating the Second Option**:
- The second option also states \( A - B = B - A \).
- This is the same as the first option and is also **incorrect** for the same reason.

**Hint**: Check the properties of subtraction in matrices; it behaves similarly to subtraction in numbers.

4. **Evaluating the Third Option**:
- The third option states that \( A - B = - (B - A) \).
- This is true because:
\[ A - B = A + (-B) = - (B + (-A)) = - (B - A) \]
- Therefore, the third option is **correct**.

**Hint**: Use the property of additive inverses to confirm the relationship between \( A - B \) and \( B - A \).

5. **Evaluating the Fourth Option**:
- The fourth option states \( A - B = B - A \).
- This is the same as the first and second options and is also **incorrect**.

**Hint**: Remember that the order of subtraction matters in matrices, just like in regular arithmetic.

### Conclusion:
The only correct statement among the options is the third one: \( A - B = - (B - A) \).