Let A and B be two nonsingular square matrices, `A^(T)` and `B^(T)` are the tranpose matrices of A and B, respectively, then which of the following are correct ?

Options

A. `B^(T)AB` is symmetric matrix if A is symmetric
B. `B^(T)AB` is symmetric matrix if B is symmetric
C. `B^(T)AB` is skew-symmetric matrix for every matrix A
D. `B^(T)AB` is skew-symmetric matrix if A is skew-symmetric

Explanation

To solve the problem, we need to analyze the expression \( B^T A B \) and determine under what conditions it is symmetric or skew-symmetric based on the properties of matrices \( A \) and \( B \).

### Step-by-Step Solution

1. **Understanding Symmetric and Skew-Symmetric Matrices**:
- A matrix \( M \) is **symmetric** if \( M^T = M \).
- A matrix \( M \) is **skew-symmetric** if \( M^T = -M \).

2. **Finding the Transpose of \( B^T A B \)**:
- We need to find the transpose of the matrix \( B^T A B \).
- Using the property of transpose, we have:
\[ (B^T A B)^T = B^T (A^T) (B^T)^T \]
- Since \( (B^T)^T = B \), we can simplify this to:
\[ (B^T A B)^T = B^T A^T B \]

3. **Condition for Symmetry**:
- For \( B^T A B \) to be symmetric, we need:
\[ B^T A B = B^T A^T B \]
- This simplifies to:
\[ A = A^T \]
- Thus, \( B^T A B \) is symmetric if \( A \) is symmetric.

4. **Condition for Skew-Symmetry**:
- For \( B^T A B \) to be skew-symmetric, we need:
\[ B^T A B = -B^T A^T B \]
- This simplifies to:
\[ A = -A^T \]
- Thus, \( B^T A B \) is skew-symmetric if \( A \) is skew-symmetric.

5. **Conclusion**:
- From the analysis, we conclude:
- \( B^T A B \) is symmetric if \( A \) is symmetric (Option 1 is correct).
- \( B^T A B \) is skew-symmetric if \( A \) is skew-symmetric (Option 4 is correct).

### Final Answer:
The correct options are:
- Option 1: \( B^T A B \) is symmetric if \( A \) is symmetric.
- Option 4: \( B^T A B \) is skew-symmetric if \( A \) is skew-symmetric.