If A and B are symmetric matrices of the same order and X = AB + BA, Y = AB - BA then (XY)<sup>T</sup> is equal to -
Step-by-Step Solution
Key Concept: Use the properties of transpose: (AB)^T = B^T A^T and (A+B)^T = A^T + B^T. Since A and B are symmetric, A^T = A and B^T = B. Calculate X^T and Y^T, then find (XY)^T = Y^T X^T.
Given A<sup>T</sup> = A and B<sup>T</sup> = B. X = AB + BA. X<sup>T</sup> = (AB + BA)<sup>T</sup> = (AB)<sup>T</sup> + (BA)<sup>T</sup> = B<sup>T</sup>A<sup>T</sup> + A<sup>T</sup>B<sup>T</sup> = BA + AB = X. So X is symmetric. Y = AB - BA. Y<sup>T</sup> = (AB - BA)<sup>T</sup> = (AB)<sup>T</sup> - (BA)<sup>T</sup> = B<sup>T</sup>A<sup>T</sup> - A<sup>T</sup>B<sup>T</sup> = BA - AB = -(AB - BA) = -Y. So Y is skew-symmetric. Now (XY)<sup>T</sup> = Y<sup>T</sup>X<sup>T</sup> = (-Y)(X) = -YX.
Correct Answer: 3