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: \( AB + BA = X \) and \( AB - BA = Y \).
Step 2: Find \( X^T \) and \( Y^T \). Given that A and B are symmetric matrices, \( A^T = A \) and \( B^T = B \). This leads to \( X^T = X \) (X is symmetric) and \( Y^T = -Y \) (Y is skew-symmetric).
Step 3: Compute \( XY \). \( XY = (AB + BA)(AB - BA) = (AB)^2 - (BA)^2 \).
Step 4: Compute \( (XY)^T \). \( (XY)^T = ((AB)^2 - (BA)^2)^T = (BA)^2 - (AB)^2 = -(XY) \).
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 \( YX \). It is observed that \( (XY)^T = -YX \).