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If the matrices, A, B and (A+B) are non-...

If the matrices, A, B and `(A+B)` are non-singular, then prove that `[A(A+B)^(-1) B]^(-1) =B^(-1)+A^(-1)`.

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`[A(A+B)^(-1) B]^(-1) =B^(-1) ((A+B)^(-1))^(-1) A^(-1)`
`=B^(-1) (A+B)A^(-1)`
`=(B^(-1) A+B^(-1)B) A^(-1)`
`=(B^(-1)A+I)A^(-1)`
`=B^(-1) A A^(-1)+IA^(-1)`
`=B^(-1)I+A^(-1)`
`=B^(-1)+A^(-1)`
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