If A and B are two non singular matrices and both are symmetric and commute each other then Options: A . &nbsp Both A − 1 B and A − 1 B − 1 are symmetric B . &nbsp A − 1 B is symmetric but A − 1 B − 1 is not symmetric C . &nbsp A − 1 B − 1 is symmetric but A − 1 B is not symmetric D . &nbsp Neither A − 1 B nor A − 1 B − 1 are symmetric

Answer: Option A : A AB =BA Previous & past multiplying both sides by A − 1 . A − 1 ( A B ) A − 1 = A − 1 ( B A ) A − 1 ( A − 1 A ) ( B A − 1 ) = A − 1 B ( A A − 1 ) ⇒ ( B A − 1 ) 1 = ( A − 1 B ) 1 = ( A − 1 ) 1 B 1 ( r e v e r s a l l a w s ) = A − 1 B ( a s B = B 1 ) ( A − 1 ) 1 = A − 1 ⇒ A − 1 B is symmetric Similarly for A − 1 B − 1 .

If A and B are two non singular matrices and both are symmetric and commute ea

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Question -

If A and B are two non singular matrices and both are symmetric and commute each other then

Options:

A . Both

A^{- 1} B

and

A^{- 1} B^{- 1}

are symmetric

B .

A^{- 1} B

is symmetric but

A^{- 1} B^{- 1}

is not symmetric

C .

A^{- 1} B^{- 1}

is symmetric but

A^{- 1} B

is not symmetric

D . Neither

A^{- 1} B

nor

A^{- 1} B^{- 1}

are symmetric

Answer: Option A
:
A
AB =BA
Previous & past multiplying both sides by

A^{- 1}

A^{- 1} (A B) A^{- 1} = A^{- 1} (B A) A^{- 1} (A^{- 1} A) (B A^{- 1}) = A^{- 1} B (A A^{- 1}) \Rightarrow (B A^{- 1})^{1} = (A^{- 1} B)^{1} = (A^{- 1})^{1} B^{1} (r e v e r s a l l a w s) = A^{- 1} B (a s B = B^{1}) (A^{- 1})^{1} = A^{- 1} \Rightarrow A^{- 1} B

is symmetric
Similarly for

A^{- 1} B^{- 1}

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