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The image presents a multiple-choice question asking which of the given statements regarding two \( n \times n \) matrices \( A \) and \( B \) are correct. Let me go through each statement and determine its correctness:

1. **\( (-A)^T = -A^T \)**  
   This is a true statement. The transpose of the negative of a matrix is the negative of the transpose of the matrix.  
   \[
   (-A)^T = -A^T
   \]
   So, this statement is **correct**.

2. **\( (AB)^T = (BA)^T \)**  
   This is **false**. The transpose of a product of matrices follows the rule:
   \[
   (AB)^T = B^T A^T
   \]
   In general, \( (AB)^T \neq (BA)^T \), unless \( A = B \), which is not assumed here.

3. **\( (A + B)^T = A^T + B \)**  
   This is **false**. The correct rule for the transpose of a sum is:
   \[
   (A + B)^T = A^T + B^T
   \]
   The statement only includes \( A^T + B \), which is incorrect.

4. **\( (AB)^T = A^T B^T \)**  
   This is **false**. As mentioned earlier, the transpose of a product of matrices is given by:
   \[
   (AB)^T = B^T A^T
   \]
   The given statement incorrectly asserts that \( (AB)^T = A^T B^T \).

### Summary of Correctness:
- 1st statement: Correct
- 2nd statement: False
- 3rd statement: False
- 4th statement: False

Would you like more details on any of these rules or further explanations?

Here are 5 related questions for further practice:
1. What is the transpose of the product of three matrices \( (ABC)^T \)?
2. How do you compute the transpose of a scalar multiple of a matrix?
3. If \( A \) is symmetric, what is \( A^T \)?
4. Can the product of two symmetric matrices be symmetric?
5. What is the condition for \( AB = BA \) to hold true for matrices \( A \) and \( B \)?

**Tip**: For matrix operations, always remember that the transpose operation reverses the order of multiplication.

Which of the following statements are correct regarding two n × n matrices A and B?

This problem tests understanding of matrix transpose properties. Evaluate which statements about two n × n matrices A and B are correct. Covers key concepts like the transpose of a sum and the product of matrices. Suitable for university-level students.

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