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

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`

C

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

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
`(A(A+B)^(-1)B)^(-1)`
`B^(-1)(A+B)A^(-1) [:' (ABC)^(-1)=C^(-1)B^(-1)A^(-1)]`
`=B^(-1)A A^(-1)+B^(-1)BA^(-1)`
`=B^(-1)+IA^(-1)=B^(-1)+A^(-1)`
`:. (A(A+B)^(-1)B)^(-1)(AB)=(B^(-1)+A^(-1))AB`
`=B^(-1)(AB)+A^(-1)(AB)`
`=B^(-1)(BA)+A^(-1)(AB)[:' AB=BA]`
`=(B^(-1)B)A+(A^(-1)A)B`
`=IB+IB=A+B`.
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