If A and B are symmetric matrices of same order, then AB - BA is a :

Options

A. Skew - symmetric matrix
B. Symmetric matrix
C. Zero matrix
D. Identity matrix

Answer

To solve the problem, we need to determine the nature of the matrix \( AB - BA \) when \( A \) and \( B \) are symmetric matrices of the same order.

### Step-by-Step Solution:

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

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

3. **Using the Property of Transpose**:
- The property of the transpose of a product states that \( (XY)^T = Y^T X^T \). Applying this, we have:
\[ (AB)^T = B^T A^T \quad \text{and} \quad (BA)^T = A^T B^T \]
- Therefore, we can rewrite the transpose as:
\[ (AB - BA)^T = B^T A^T - A^T B^T \]

4. **Substituting the Symmetric Properties**:
- Since \( A \) and \( B \) are symmetric, we can replace \( A^T \) with \( A \) and \( B^T \) with \( B \):
\[ (AB - BA)^T = B A - A B \]
- This can be rearranged as:
\[ (AB - BA)^T = - (AB - BA) \]

5. **Conclusion**:
- The equation \( (AB - BA)^T = - (AB - BA) \) indicates that \( AB - BA \) is skew-symmetric. A matrix \( C \) is skew-symmetric if \( C^T = -C \).

### Final Answer:
Thus, \( AB - BA \) is a skew-symmetric matrix.