If A and B symmetric matrices of the same order then AB-BA is a matrix which is

Options

A. null
B. unit
C. symmetric
D. skew symmetric

Answer

To solve the problem, we need to show that if A and B are symmetric matrices of the same order, then the matrix \( AB - BA \) is skew-symmetric.

### Step-by-Step Solution:

1. **Definition of Symmetric Matrices**:
A matrix \( A \) is symmetric if \( A = A^T \) (where \( A^T \) is the transpose of \( A \)). Similarly, for matrix \( B \), we have \( B = B^T \).

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

3. **Applying Transpose to Each Term**:
Now, apply the transpose to each product:
\[ (AB)^T = B^T A^T \quad \text{and} \quad (BA)^T = A^T B^T \]
Therefore, we can rewrite the expression:
\[ (AB - BA)^T = B^T A^T - A^T B^T \]

4. **Substituting Symmetric Properties**:
Since \( A \) and \( B \) are symmetric, we know:
\[ B^T = B \quad \text{and} \quad A^T = A \]
Substituting these into our expression gives:
\[ (AB - BA)^T = BA - AB \]

5. **Rearranging the Terms**:
Notice that:
\[ BA - AB = -(AB - BA) \]
This means:
\[ (AB - BA)^T = -(AB - BA) \]

6. **Conclusion**:
The property \( (AB - BA)^T = -(AB - BA) \) indicates that \( AB - BA \) is skew-symmetric.

### Final Result:
Thus, if \( A \) and \( B \) are symmetric matrices of the same order, then \( AB - BA \) is a skew-symmetric matrix.