Linear Algebra Proof using induction

I'm having trouble starting because I haven't done induction in a long time. The problem says:

Let A be an n x n invertible matrix. Prove that for all m in the naturals, A^{m }is invertible and determine a formula for its inverse in terms of A^{-1}. Hint: use induction. My professor told us to only do this for m=2 and m=3.

Any help would be greatly appreciated!

Re: Linear Algebra Proof using induction

For the induction step use

$\displaystyle (A^{k})(A^k)^{-1}=I \Leftrightarrow A(A^k)(A^k)^{-1}=A \Leftrightarrow (A^{k+1})(A^k)^{-1}A^{-1}=AA^{-1}=I$

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