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

Options

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

Answer

To determine whether \( AB - BA \) is a symmetric or skew-symmetric matrix when \( A \) and \( B \) are symmetric matrices of the same order, we can follow these steps:

### Step 1: Understand the properties of symmetric matrices
A matrix \( A \) is symmetric if \( A = A^T \) (the transpose of \( A \) is equal to \( A \)). Similarly, for matrix \( B \), we have \( B = B^T \).

**Hint:** Remember that symmetric matrices have the property that their transpose is equal to themselves.

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

**Hint:** Use the property that the transpose of a product of matrices is the product of their transposes in reverse order.

### Step 3: Apply the transpose property to each term
Now, applying the transpose property to each term:
\[ (AB)^T = B^T A^T \quad \text{and} \quad (BA)^T = A^T B^T \]
Substituting these into our expression gives:
\[ (AB - BA)^T = B^T A^T - A^T B^T \]

**Hint:** Remember to reverse the order of multiplication when taking the transpose of a product.

### Step 4: Substitute the symmetric properties of \( A \) and \( B \)
Since \( A \) and \( B \) are symmetric, we can replace \( A^T \) with \( A \) and \( B^T \) with \( B \):
\[ (AB - BA)^T = B A - A B \]

**Hint:** Use the fact that for symmetric matrices, the transpose of the matrix is the matrix itself.

### Step 5: Rearranging the terms
Notice that \( B A - A B = - (A B - B A) \). Therefore, we can write:
\[ (AB - BA)^T = - (AB - BA) \]

**Hint:** Recognize that this shows a specific relationship between the matrix and its transpose.

### Step 6: Conclusion
The equation \( (AB - BA)^T = - (AB - BA) \) indicates that \( AB - BA \) is skew-symmetric. A matrix \( Z \) is skew-symmetric if \( Z^T = -Z \).

Thus, we conclude that \( AB - BA \) is a skew-symmetric matrix.

**Final Answer:** \( AB - BA \) is a skew-symmetric matrix.