Fill in the blanks in each of the

If A and B are symmetric matrices, then

(i) AB – BA is a _________.

(ii) BA – 2AB is a _________.

(i) AB – BA is a Skew Symmetric matrix

Given A’=A and B’=B

⇒ (AB-BA)’=(AB)’-(BA)’

⇒ (AB)’-(BA)’=B’A’-A’B’

⇒ B’A’-A’B’=BA-AB=-(AB-BA)

⇒ (AB-BA)’=-(AB-BA) (skew symmetric matrix)

Eg. Let A =

B=

⇒ AB= and BA=

⇒ AB-BA=

⇒ (AB-BA)’=

⇒ -(AB-BA)=

(ii) BA – 2AB is a Neither Symmetric nor Skew Symmetric matrix

Given A’=A and B’=B

⇒ (BA-2AB)’=(BA)’-(2AB)’

⇒ (BA)’-(2AB)’=A’B’-2B’A’

⇒ A’B’-2B’A’=AB-2BA=-(2BA-AB)

⇒ (BA-2AB)’=-(2BA-AB)

Eg. Let A =

B=

⇒ AB= and BA=

⇒ BA-2AB=