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If A and B are symmetric non-singular ma...

If A and B are symmetric non-singular matrices of same order,AB = BA and `A^(-1)B^(-1)` exist, prove that `A^(-1)B^(-1)` is symmetric.

Text Solution

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`because" " A' = A, B'= B and |A| ne 0, |B|' ne 0`
` therefore" " (A^(-1)B^(-1)'(A^(-1)'`
` =(B')^(-1) (A')^(-1)`
`= B^(-1) A^(-1) " " [because A' = A and B' = B]`
`=(AB)^(-1)`
` =(BA)^(-1) " "[because AB = BA]`
`= A^(-1) B^(-1)`
Hence,` A^(-1) B^(-1)` is symmetric.
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