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Showing A Matrix Property Is True

  1. Oct 19, 2012 #1
    1. The problem statement, all variables and given/known data
    Let A be an invertible matrix. Show that [itex](A^n)^{-1} = (A^{-1})^n[/itex]


    2. Relevant equations



    3. The attempt at a solution
    I want to begin on the left side of the equality sign; but I am having a little difficulty on expanding it. I started to--[itex](A^n)^{-1} = AAA...A^{-1}[/itex]--but it just didn't appear correct. Could someone help me?
     
  2. jcsd
  3. Oct 19, 2012 #2

    HallsofIvy

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    I would be inclined to prove it using "induction". If n= 1, this just says that [itex](A)^{-1}= A^{-1}[/itex]. Now, suppose [itex](A^k)^{-1}= (A^{-1})^k[/itex]. What can you say about [itex](A^{k+1})^{-1}= ((A^k)A)^{-1}[/itex]?

    (You will need to prove separately that [itex](AB)^{-1}= B^{-1}A^{-1}[/itex].)

    By the way, you have
    .
    No, that is not correct. (An)-1 is the multiplicative inverse of An: (An)(An)-1= I.
     
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