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Math Help - Linear Algebra Proof using induction

  1. #1
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    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, Am 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!
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  2. #2
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    Re: Linear Algebra Proof using induction

    For the induction step use
    (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

    I leave the details for you to fill in.
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