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: Recall what symmetric means.
Since $ A $ and $ B $ are symmetric matrices, we have $ A^T = A $ and $ B^T = B $. This is our key tool.
Step 2: Find the transpose of X.
Given $ X = AB + BA $. Take the transpose of both sides: $ X^T = (AB + BA)^T = (AB)^T + (BA)^T = B^T A^T + A^T B^T $. Since $ A^T = A $ and $ B^T = B $, this becomes $ X^T = BA + AB = X $. So $ X $ is symmetric.
Step 3: Find the transpose of Y.
Given $ Y = AB - BA $. Take the transpose: $ Y^T = (AB - BA)^T = B^T A^T - A^T B^T = BA - AB = -(AB - BA) = -Y $. So $ Y $ is skew-symmetric.
Step 4: Apply the transpose to XY.
We need $ (XY)^T $. Using the property $ (PQ)^T = Q^T P^T $: \[ (XY)^T = Y^T X^T \]
Step 5: Substitute the results.
We found $ X^T = X $ and $ Y^T = -Y $. So: \[ (XY)^T = (-Y)(X) = -YX \]
Step 6: State the final answer.
\[ \boxed{(XY)^T = -YX} \]