Assertion (A) : If A and B are symmetric matrices, then AB - BA is a skew symmetric matrix
Reason (R) : (AB)' = B' A'

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is true

Answer

To solve the assertion and reason problem, we need to prove that if \( A \) and \( B \) are symmetric matrices, then \( AB - BA \) is a skew-symmetric matrix. We will also verify the reason provided.

### Step-by-Step Solution:

1. **Definition of Symmetric Matrices**:
- A matrix \( A \) is symmetric if \( A = A^T \).
- A matrix \( B \) is symmetric if \( B = B^T \).

2. **Definition of Skew-Symmetric Matrices**:
- A matrix \( C \) is skew-symmetric if \( C = -C^T \).

3. **Expression to Analyze**:
- We need to analyze the expression \( AB - BA \).

4. **Taking the Transpose**:
- We start by taking the transpose of the expression \( AB - BA \):
\[ (AB - BA)^T = (AB)^T - (BA)^T \]

5. **Using the Property of Transpose**:
- The property of transpose states that \( (XY)^T = Y^T X^T \). Thus, we can write:
\[ (AB)^T = B^T A^T \quad \text{and} \quad (BA)^T = A^T B^T \]
- Substituting these into our expression gives:
\[ (AB - BA)^T = B^T A^T - A^T B^T \]

6. **Substituting for Symmetric Matrices**:
- Since \( A \) and \( B \) are symmetric, we have \( A^T = A \) and \( B^T = B \). Therefore:
\[ (AB - BA)^T = B A - A B \]

7. **Rearranging the Expression**:
- We can rewrite the expression as:
\[ (AB - BA)^T = - (AB - BA) \]

8. **Conclusion**:
- This shows that \( AB - BA \) is skew-symmetric since it satisfies the condition \( (AB - BA)^T = - (AB - BA) \).

9. **Verifying the Reason**:
- The reason states that \( (AB)^T = B^T A^T \). This is indeed true due to the property of transposes. Since both \( A \) and \( B \) are symmetric, the reason is valid.

### Final Conclusion:
Both the assertion (A) and the reason (R) are true, and the reason (R) is the correct explanation of the assertion (A).