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If A and B are symmetric matrices, prove that AB-BA is a skew

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If $A$ and $B$ are symmetric matrices, prove that $A B - B A$ is a skew symmetric matrix.

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A

and

B

are symmetric matrices, prove that

A B - B A

is a skew symmetric matrix.

8 mins ago

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Text solutionVerified

It is given that

A

and

B

are symmetric matrices.

Therefore, we have:

A^{'} = A

and

B^{'} = B

..............(1)

Now,

(A B - B A)^{'} = (A B)^{'} - (B A)^{'}, [∵ (A - B)^{'} = A^{'} - B^{'}]

= B^{'} A^{'} - A^{'} B^{'}, [∵ (A B)^{'} = B^{'} A^{'}]

= B A - A B

................ [using (1) ]

= - (A B - B A)

∴ (A B - B A)^{'} = - (A B - B A)

Thus,

(A B - B A)

is a skew-symmetric matrix.

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Question Text	If $A$ and $B$ are symmetric matrices, prove that $A B - B A$ is a skew symmetric matrix.
Answer Type	Text solution:1
Upvotes	150