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=
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