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A^k matrix singularity and (A^k)^-1 = (A^-1)^k

  1. Jul 13, 2009 #1
    1. The problem statement, all variables and given/known data

    Let A be nonsingular. Prove That for any positive integer k , A^k is nonsingular, And (A^k)^-1 = (A^-1)^k.

    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Jul 13, 2009
  2. jcsd
  3. Jul 13, 2009 #2
    What have you tried so far?
     
  4. Jul 13, 2009 #3

    statdad

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    Fill this in: a problem that requires you to prove a simple expression is true for every positive integer [tex] k [/tex] is a good candidate for m ************ ******n
     
  5. Jul 13, 2009 #4
    You answer was not complete ...What are the * ?
    Please somebody help me!
     
  6. Jul 13, 2009 #5

    Office_Shredder

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    Start small. Can you prove it's true for k=2? How can you generalize the proof?
     
  7. Jul 13, 2009 #6
    I can't prove it for 2 ,,,don't know How to generalize that:cry:
     
  8. Jul 13, 2009 #7

    Office_Shredder

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    Start by contradiction. If A is nonsingular, then if A2 is singular what can we prove about A? Try messing around with the equation A2v = 0
     
  9. Jul 13, 2009 #8

    HallsofIvy

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    Then what do you know? Under what conditions is a matrix "singular"? What does having an inverse mean?
     
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