UPSC NDA Math Model Paper 6

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=(A)T−(B)T
(PQ−QP)T=(PQ)T−(QP)T
Using again property of transpose, (AB)T=(B)T(A)T
(PQ)T−(QP)T=(Q)T(P)T−(P)T(Q)T ...........(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.