Need help proving this matrices equation

[nickgar712]

We are given matrices A, B with n x n dimensions. We know that A^-1 ,B^-1,(A-B)^-1,(B^-1 - A^-1)^-1 do exist. Prove:

(A-B)^-1 = A^-1 + A^-1*(B^-1 - A^-1)^-1*A^-1

 

[Deleted member 4993]

We are given matrices A, B with n x n dimensions. We know that A^-1 ,B^-1,(A-B)^-1,(B^-1 - A^-1)^-1 do exist. Prove:

(A-B)^-1 = A^-1 + A^-1*(B^-1 - A^-1)^-1*A^-1

Suppose A and B were "numbers" - instead of matrices. Would you expect the given relation to hold good? Why or why not?

Please show us what you have tried and exactly where you are stuck.

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