Question

if P and Q are symmetric matrices of the same order then PQ-QP​

Answer 1

Answer:

(PQ - QP ) is a Skew Symmetric Matrix

Step-by-step explanation:

Since:

P and Q Symmetric Matrices therefore

Transport of P = T(P) = P                  .....(1)

Transport of Q = T(Q) = Q               .....(2)

Now

Transport of (PQ -QP) = T( PQ - QP )  

Using the property of Transport   T(A-B)=T(A) - T(B)

T( PQ - QP ) = T(PQ) - T(QP)                            

Using again property of transport     T(AB) = T(B) T(A)

T(PQ) - T(QP) = T(Q) T(P) - T(P) T(Q)        ............(3)

Using Equations (1) and (2) in (3) we get

T(PQ) - T(QP) = QP - PQ

T(PQ) - T(QP) = - ( PQ - QP)                                  

SO  

Transport of (PQ - QP) = T( PQ - QP ) = - ( PQ - QP)

Which show that  

(PQ - QP ) is a Skew Symmetric Matrix.

Answer 2

Answer:

PQ-QP is a  skew symmetric matrix

Step-by-step explanation:

Concept used:

Symmetric matrix:

A square matrix A is said to be symmetric if $A^T=A$

A square matrix A is said to be skew symmetric if $A^T=-A$

If A and B be two square matrices of same order, then

$(AB)^T=B^TA^T$

Given:

P and Q are symmetric matrices.

Then,

$P=P^T$...........(1)

$Q=Q^T$.............(2)

Now,

$(PQ-QP)^T$

$=(PQ)^T-(QP)^T$

$=Q^TP^T-P^TQ^T$

$=QP-PQ$     (using (1) and (2))

$=-(PQ-QP)$

$(PQ-QP)^T=-(PQ-QP)$

Hence PQ-QP is skew symmetric matrix.