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Fill in the blanks:

If A and B are symmetric matrices, then 

(i) AB – BA is a _________. 

(ii) BA – 2AB is a _________.

1 Answer

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Best answer

(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)

(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)

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