A and B are square symmetric matrices of the same order.
For a symmetric matrix, AT=A (the matrix equals its own transpose).
Since A and B are symmetric:
AT=A
BT=B
Let P=AB−BA
Taking the transpose:
PT=(AB−BA)T
PT=(AB)T−(BA)T
Using the transpose property (XY)T=YTXT:
PT=BTAT−ATBT
Substituting AT=A and BT=B:
PT=BA−AB
PT=−(AB−BA)
PT=−P
A matrix M is skew-symmetric when MT=−M.
Since PT=−P, the matrix (AB−BA) is a skew-symmetric matrix.