SOLUTION: proof : A^n is invertible and A^n.-1 = A^-n .

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Question 1109397: proof :
A^n is invertible and A^n.-1 = A^-n .
Answer by ikleyn(53763) About Me  (Show Source):
You can put this solution on YOUR website!
.
Since your formulation is far from to be perfect, I will edit the formulation:


        Prove:
        If A is invertible matrix, then %28A%5En%29%5E%28-1%29 = A%5E%28-n%29. 

I will prove it for n = 2, and you will see from there how to do it in the general case.

We are given that  A%2AA%5E%28-1%29 = I.


Then  A%5E2%2AA%5E%28-2%29 = A%2AA%2A%28A%5E%28-1%29%29%2A%28A%5E%28-1%29%29 = A%2A%28A%2A%28A%5E%28-1%29%29%29%2A%28A%5E%28-1%29%29 = A%2AI%2A%28A%5E%28-1%29%29 = A%2A%28A%5E%28-1%29%29 = I.


QED.