Question

If \(A\) and \(B\) are symmetric matrices of 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:

For symmetric matrices: \[ A^T=A \] and for skew-symmetric matrices: \[ A^T=-A \] Also remember: \[ (AB)^T=B^TA^T \] which is frequently used in matrix algebra problems.

Answer 1

The Correct Option is C

Step 1: Use the property of symmetric matrices.
Since \(A\) and \(B\) are symmetric matrices, we have \[ A^T=A \] and \[ B^T=B \] Given \[ X=AB+BA \] Taking transpose on both sides, \[ X^T=(AB+BA)^T \] Using \[ (AB)^T=B^TA^T, \] we get \[ X^T=B^TA^T+A^TB^T \] Since \(A^T=A\) and \(B^T=B\), \[ X^T=BA+AB \] Hence, \[ X^T=X \] Therefore, \(X\) is symmetric.

Step 2: Find the transpose of \(Y\).
Given \[ Y=AB-BA \] Taking transpose, \[ Y^T=(AB-BA)^T \] So, \[ Y^T=B^TA^T-A^TB^T \] Using symmetry of \(A\) and \(B\), \[ Y^T=BA-AB \] Thus, \[ Y^T=-(AB-BA) \] Hence, \[ Y^T=-Y \] Therefore, \(Y\) is skew-symmetric.

Step 3: Find \((XY)^T\).
Using the transpose property, \[ (XY)^T=Y^TX^T \] Since \[ Y^T=-Y \] and \[ X^T=X, \] we get \[ (XY)^T=(-Y)(X) \] Therefore, \[ (XY)^T=-YX \]

Step 4: Final conclusion.
Hence, \[ \boxed{(XY)^T=-YX} \]