Which of the following is/are correct ?

Options

A. B' AB is symmetric if A is symmetric
B. B' AB is skew-symmetric if A is symmetric
C. B' AB is symmetric if A is skew-symmetric
D. None of these

Answer

To determine which of the given options is correct regarding matrices, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to check the correctness of the statements involving matrices A and B, where A is symmetric and B is skew-symmetric.

2. **Option A**: We need to check if \( B^T A B \) is symmetric.
- **Definition**: A matrix \( M \) is symmetric if \( M^T = M \).
- **Transposing \( B^T A B \)**:
\[ (B^T A B)^T = B^T A^T (B^T)^T = B^T A^T B \]
- Since \( A \) is symmetric, \( A^T = A \):
\[ (B^T A B)^T = B^T A B \]
- Thus, \( B^T A B \) is symmetric. **Option A is correct**.

3. **Option B**: We need to check if \( B^T A B \) is skew-symmetric.
- **Definition**: A matrix \( N \) is skew-symmetric if \( N^T = -N \).
- Using the same transposition:
\[ (B^T A B)^T = B^T A B \]
- Since we found that \( (B^T A B)^T = B^T A B \), it cannot be skew-symmetric because it does not equal \(-B^T A B\). **Option B is incorrect**.

4. **Option C**: We need to check if \( B^T A B \) is neither symmetric nor skew-symmetric.
- Since we have already established that \( B^T A B \) is symmetric, it cannot be neither. **Option C is incorrect**.

### Conclusion:
- **Correct Option**: Only **Option A** is correct.