P and Q Symmetric Matrices therefore
Transpose of P = P ..... (1)
Transpose of Q = Q ..... (2)
Now
Transpose of
(PQ−QP)=(PQ−QP)T Using the property of Transpose,
(A−B)T=AT−BT (PQ−QP)T=(PQ)T−(QP)T Using again property of transpose,
(AB)T=BTAT (PQ)T−(QP)T=QTPT−PTQT ...........(3)
Using Equations (1) and (2) in (3) we get,
(PQ)T−(QP)T=QP−PQ (PQ)T−(QP)T=−(PQ−QP) So,
Transpose of
(PQ−QP)=(PQ−QP)T=−(PQ−QP) Which show that
(PQ−QP) is a Skew Symmetric Matrix.
Hence, option (4) is correct.