if `A` and `B` are symmetric matrices of same order then `(AB-BA)` is :

Options

A. Zero matrix
B. identity matrix
C. skew symmetric matrix
D. symmetric matrix

Explanation

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. **Understand the Properties of Symmetric Matrices**:
A matrix \( M \) is symmetric if \( M^T = M \). Given that both \( A \) and \( B \) are symmetric, we have:
\[ A^T = A \quad \text{and} \quad B^T = B \]

2. **Take the Transpose of \( AB - BA \)**:
We need to find the transpose of the expression \( AB - BA \):
\[ (AB - BA)^T = (AB)^T - (BA)^T \]

3. **Apply the Transpose Property**:
Using the property of the transpose of a product of matrices, we have:
\[ (AB)^T = B^T A^T \quad \text{and} \quad (BA)^T = A^T B^T \]
Substituting the symmetric properties:
\[ (AB)^T = B A \quad \text{and} \quad (BA)^T = A B \]

4. **Substituting Back**:
Now substituting these back into our expression:
\[ (AB - BA)^T = BA - AB \]

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

6. **Conclusion about the Nature of the Matrix**:
The equation \( (AB - BA)^T = -(AB - BA) \) indicates that \( AB - BA \) is skew-symmetric. A matrix \( M \) is skew-symmetric if \( M^T = -M \).

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