If A and B are symmetric matrices of the same order and X = AB + BA and Y = AB - BA, then (XY)T is equal to
-YX
Given that A and B are symmetric.
AT=A, BT=B
X=AB+BA⇒XT=(AB)T+(BA)T=BTAT+ATBT=BA+AB=X, ((AB)T=BTAT)Y=AB−BA⇒YT=(AB)T−(BA)T=BTAT−ATBT=BA−AB=−Y, ((AB)T=BTAT)Now (XY)T=YTXT=−YX