If A,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**:
By definition, a matrix \( A \) is symmetric if \( A^T = A \) and a matrix \( B \) is symmetric if \( B^T = B \).

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

3. **Applying the Transpose Property**:
Using the property of transpose, we have:
\[ (AB - BA)^T = (AB)^T - (BA)^T \]

4. **Using the Product of Transposes**:
The transpose of a product of matrices follows the rule \( (XY)^T = Y^T X^T \). Therefore:
\[ (AB)^T = B^T A^T \quad \text{and} \quad (BA)^T = A^T B^T \]

5. **Substituting the Transposes**:
Substituting these into our expression gives:
\[ (AB - BA)^T = B^T A^T - A^T B^T \]

6. **Using the Symmetry of A and B**:
Since \( A \) and \( B \) are symmetric, we can replace \( A^T \) with \( A \) and \( B^T \) with \( B \):
\[ (AB - BA)^T = BA - AB \]

7. **Rearranging the Expression**:
Notice that:
\[ BA - AB = -(AB - BA) \]
Therefore:
\[ (AB - BA)^T = -(AB - BA) \]

8. **Conclusion**:
The expression \( AB - BA \) is equal to its own negative transpose, which indicates that \( AB - BA \) is a skew-symmetric matrix.

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

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