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 \( AB - BA \) given that \( A \) and \( B \) are symmetric matrices of the same order.

### Step-by-Step Solution:

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

2. **Finding the Transpose of \( AB - BA \)**:
- We need to compute the transpose of the expression \( AB - BA \).
- Using the property of transposes, we have:
\[ (AB - BA)^T = (AB)^T - (BA)^T \]

3. **Applying the Transpose Property**:
- The transpose of a product of matrices follows the rule \( (XY)^T = Y^T X^T \).
- Therefore, we can write:
\[ (AB)^T = B^T A^T \quad \text{and} \quad (BA)^T = A^T B^T \]

4. **Substituting the Symmetric Properties**:
- Since \( A \) and \( B \) are symmetric, we know \( A^T = A \) and \( B^T = B \).
- Substituting these into our expression gives:
\[ (AB - BA)^T = B A - A B \]
- This can be rewritten as:
\[ (AB - BA)^T = - (AB - BA) \]

5. **Conclusion about the Nature of \( AB - BA \)**:
- 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.