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If A and B are two non-singular matrice...

If `A ` and `B` are two non-singular matrices which commute, then `(A(A+B)^(-1)B)^(-1)(AB)=`

A

`A+B`

B

`A^(-1)+B^(-1)`

C

`A^(-1)+B`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` `(A(A+B)^(-1)B)^(-1)(AB)`
`=B^(-1)(A+B^(-1))^(-1)A^(-1)(AB)`
`=B^(-1)(A+B)A^(-1)(AB)`
`=(B^(-1)A+I)A^(-1)(AB)`
`=(B^(-1)A A^(-1)+A^(-1))(AB)`
`=(B^(-1)+A^(-1))(AB)`
`=B^(-1)AB+A^(-1)AB`
`=B^(-1)BA+A^(-1)AB`
`=A+B`
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