Question

If A and B are symmetric matrices of the same order such that \( AB + BA = X \) and \( AB - BA = Y \), then \( (XY)^T = \)

• A. \( XY \)
• B. \( X^TY^T \)
• C. \( -YX \)
• D. \( -Y^TX^T \)

Hint:

Remember that \( (AB)^T = B^TA^T \), and if A is symmetric, \( A^T = A \).

Answer 1

The Correct Option is C

Step 1: {Use the given equations}
We are given \( AB + BA = X \) and \( AB - BA = Y \). 
Step 2: {Find \( X^T \) and \( Y^T \)}
Since A and B are symmetric matrices, \( A^T = A \) and \( B^T = B \). \[ X^T = (AB + BA)^T = (AB)^T + (BA)^T = B^TA^T + A^TB^T = BA + AB = X \] \[ Y^T = (AB - BA)^T = (AB)^T - (BA)^T = B^TA^T - A^TB^T = BA - AB = -(AB - BA) = -Y \] So, \( X \) is symmetric and \( Y \) is skew-symmetric. 
Step 3: {Compute \( XY \)}
\[ XY = (AB + BA)(AB - BA) = (AB)^2 - (BA)^2 \] 
Step 4: {Compute \( (XY)^T \)}
\[ (XY)^T = ((AB)^2 - (BA)^2)^T = ((AB)^2)^T - ((BA)^2)^T = (B^TA^T)^2 - (A^TB^T)^2 = (BA)^2 - (AB)^2 = -(XY) \] 
Step 5: {Compute \( YX \)}
\[ YX = (AB - BA)(AB + BA) = (AB)^2 - (BA)^2 = -(BA)^2 + (AB)^2 = -((BA)^2 - (AB)^2) = -XY \] 
Step 6: {Compare \( (XY)^T \) with \( YX \)}
We found that \( (XY)^T = -YX \) . 
 

Answer 2

Step 1: Given Conditions
We are given that \( A \) and \( B \) are symmetric matrices of the same order, and that: \[ AB + BA = X, \quad AB - BA = Y. \] We are asked to find \( (XY)^T \).

Step 2: Find \( XY \)
We first compute the product \( XY \): \[ XY = (AB + BA)(AB - BA). \] Expanding this, we get: \[ XY = AB \cdot AB - AB \cdot BA + BA \cdot AB - BA \cdot BA. \] Simplifying each term: \[ XY = A B^2 A - ABBA + BAAB - B^2 A B. \] Since \( A \) and \( B \) are symmetric matrices, we know that \( A^T = A \) and \( B^T = B \). Therefore, we can rearrange the terms accordingly. However, instead of continuing to expand and simplify, we need to focus on finding the transpose of \( XY \).

Step 3: Find the Transpose \( (XY)^T \)
The transpose of a product of matrices follows the rule: \[ (AB)^T = B^T A^T. \] Thus, we find the transpose of \( XY \): \[ (XY)^T = (AB + BA)(AB - BA)^T. \] Taking the transpose of each part: \[ (AB + BA)^T = A^T B^T + B^T A^T = AB + BA \quad (\text{since } A \text{ and } B \text{ are symmetric}). \] \[ (AB - BA)^T = A^T B^T - B^T A^T = AB - BA. \] Therefore: \[ (XY)^T = (AB + BA)(AB - BA). \] But this is the same as \( XY \), so we conclude that: \[ (XY)^T = -YX. \]

Step 4: Conclusion
Thus, the value of \( (XY)^T \) is: \[ (XY)^T = -YX. \]

The correct answer is: \( -YX \)