Consider the following statements in respect of symmetric matrices A and B
1. AB is symmetric.
2. `A^(2) + B^(2)` is symmetric.
Which of the above statement(s) is/are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To determine the correctness of the statements regarding symmetric matrices A and B, we will analyze each statement step by step.

### Step 1: Analyze the first statement - "AB is symmetric."
- A matrix \( M \) is symmetric if \( M^T = M \).
- For the product of two matrices \( AB \), we need to check if \( (AB)^T = AB \).
- Using the property of transposes, we have:
\[ (AB)^T = B^T A^T \]
- Since A and B are symmetric, we know that \( A^T = A \) and \( B^T = B \).
- Therefore, substituting these into our equation gives us:
\[ (AB)^T = B A \]
- For \( AB \) to be symmetric, we would need \( B A = AB \). However, this is not generally true for arbitrary matrices A and B.
- Hence, the first statement is **incorrect**.

### Step 2: Analyze the second statement - "A² + B² is symmetric."
- We need to check if \( A^2 + B^2 \) is symmetric.
- A matrix \( M \) is symmetric if \( M^T = M \).
- We calculate the transpose of \( A^2 + B^2 \):
\[ (A^2 + B^2)^T = A^2 + B^2 \]
- Using the property of transposes, we have:
\[ (A^2)^T + (B^2)^T \]
- Since \( A \) and \( B \) are symmetric, we know that:
\[ (A^2)^T = (A^T A^T) = A A = A^2 \]
\[ (B^2)^T = (B^T B^T) = B B = B^2 \]
- Therefore, we can conclude:
\[ (A^2 + B^2)^T = A^2 + B^2 \]
- This shows that \( A^2 + B^2 \) is symmetric.
- Hence, the second statement is **correct**.

### Conclusion:
- The first statement is incorrect, and the second statement is correct.
- Therefore, the answer is **Option B: Only statement 2 is correct**.