Let A and B two symmetric matrices of order 3.
Statement 1 : `A(BA)` and `(AB)A` are symmetric matrices.
Statement 2 : `AB` is symmetric matrix if matrix multiplication of A with B is commutative.

Options

A. Statement 1 is false, statement 2 is true.
B. Statement 1 is true, statement 2 is true, statement 2 is a correct explanation for statement 1.
C. Statement 1 is true, statement 2 is true, statement 2 is not a correct explanation for statement 1.
D. Statement 1 is true, statement 2 is false.

Explanation

To solve the problem, we need to analyze the two statements about symmetric matrices \( A \) and \( B \) of order 3.

### Step 1: Understanding Symmetric Matrices
A matrix \( A \) is symmetric if \( A = A^T \), where \( A^T \) denotes the transpose of matrix \( A \). Similarly, \( B \) is symmetric if \( B = B^T \).

### Step 2: Proving Statement 1: \( A(BA) \) and \( (AB)A \) are symmetric matrices
1. **For \( A(BA) \)**:
- We need to find the transpose of \( A(BA) \):
\[ (A(BA))^T = (BA)^T A^T \]
- Using the property of transpose, \( (XY)^T = Y^T X^T \):
\[ (BA)^T = A^T B^T \]
- Since both \( A \) and \( B \) are symmetric:
\[ (BA)^T = A B \]
- Therefore:
\[ (A(BA))^T = (AB)A \]
- Thus, \( A(BA) = (AB)A \), which shows that \( A(BA) \) is symmetric.

2. **For \( (AB)A \)**:
- We need to find the transpose of \( (AB)A \):
\[ ((AB)A)^T = A^T (AB)^T \]
- Again using the property of transpose:
\[ (AB)^T = B^T A^T \]
- Since both \( A \) and \( B \) are symmetric:
\[ (AB)^T = BA \]
- Therefore:
\[ ((AB)A)^T = A (BA) \]
- Thus, \( (AB)A = A(BA) \), which shows that \( (AB)A \) is symmetric.

### Conclusion for Statement 1:
Both \( A(BA) \) and \( (AB)A \) are symmetric matrices. Therefore, Statement 1 is **true**.

### Step 3: Proving Statement 2: \( AB \) is symmetric if \( AB = BA \)
- We need to check if \( AB \) is symmetric under the condition \( AB = BA \):
\[ (AB)^T = B^T A^T \]
- Since \( A \) and \( B \) are symmetric:
\[ (AB)^T = BA \]
- Given that \( AB = BA \):
\[ (AB)^T = AB \]
- Thus, \( AB \) is symmetric.

### Conclusion for Statement 2:
Statement 2 is also **true**.

### Final Conclusion:
Both statements are true.