If `A and B` are symmetric matrices of the same order and `X=AB+BA and Y=AB-BA,` then `(XY)^T` is equal to : (A) `XY` (B) `YX` (C) `-YX` (D) non of these

Options

A. `XY`
B. `YX`
C. `-YX`
D. none of these

Explanation

To solve the problem, we need to find \((XY)^T\) where \(X = AB + BA\) and \(Y = AB - BA\), given that \(A\) and \(B\) are symmetric matrices.

### Step-by-Step Solution:

1. **Understanding Symmetric Matrices**:
Since \(A\) and \(B\) are symmetric matrices, we have:
\[ A^T = A \quad \text{and} \quad B^T = B \]

2. **Expressing \(X\) and \(Y\)**:
We define:
\[ X = AB + BA \]
\[ Y = AB - BA \]

3. **Finding \((XY)^T\)**:
We want to find \((XY)^T\). Using the property of transpose, we have:
\[ (XY)^T = Y^T X^T \]

4. **Calculating \(Y^T\)**:
We calculate \(Y^T\):
\[ Y^T = (AB - BA)^T = (AB)^T - (BA)^T \]
Using the property \((AB)^T = B^T A^T\):
\[ Y^T = B A - A B = BA - AB \]

5. **Calculating \(X^T\)**:
Now we calculate \(X^T\):
\[ X^T = (AB + BA)^T = (AB)^T + (BA)^T \]
Again using the property \((AB)^T = B^T A^T\):
\[ X^T = BA + AB \]

6. **Substituting Back**:
Now substituting \(Y^T\) and \(X^T\) into the equation for \((XY)^T\):
\[ (XY)^T = (BA - AB)(BA + AB) \]

7. **Expanding the Product**:
We expand the product:
\[ (XY)^T = BA \cdot BA + BA \cdot AB - AB \cdot BA - AB \cdot AB \]

8. **Rearranging Terms**:
Notice that:
\[ BA \cdot AB - AB \cdot BA = (AB - BA)(AB + BA) \]
Thus:
\[ (XY)^T = - (AB + BA)(AB - BA) \]

9. **Identifying \(X\) and \(Y\)**:
We can see that:
\[ AB + BA = X \quad \text{and} \quad AB - BA = Y \]
Therefore:
\[ (XY)^T = -YX \]

### Conclusion:
Thus, we conclude that:
\[ (XY)^T = -YX \]
The correct answer is option (C) \(-YX\).