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.