If P and Q are symmetric matrices of the same order then PQ - QP is
If P and Q are symmetric matrices of the same order then PQ - QP is Correct Answer skew symmetric matrix
Concept:
If matrix P is symmetric matrix then
Transpose of P = P
Consider A and B matrices, (AB)T = (B)T(A)T
If matrix P is skew symmetric matrix then
Transpose of P = -P
Calculation:
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.
