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Math Help - prove that matrix is invertible

  1. #1
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    prove that matrix is invertible

    Hi!

    i need to prove that if matrix A^n is invertible, then A is also invertible.

    TIA!
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  2. #2
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    Re: prove that matrix is invertible

    Hey Stormey.

    Recall that det(AB) = det(A)*det(B) for operators A and B.
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  3. #3
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    Re: prove that matrix is invertible

    it's easier to use chiro's hint if you prove the contrapositive:

    if A is singular, so is An
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    Re: prove that matrix is invertible

    we didn't learn determinants yet, so i'm guessing it can be solved without using it.

    nevertheless, from what i read in wiki, if matrix A is invetrtible, then its determinant is not 0.
    and since i want to prove the contrapositive, i need to assume it's singular, and the its det is equal 0.
    so:

    det(A)=0

    i can multiply it by det(A) n-1 times, and then say that det(A^n)=0
    but what conclusion can i draw from it?
    does that prove it?

    BTW, i have no idea what determinant is, so bare with me...
    Last edited by Stormey; December 5th 2012 at 04:06 AM.
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  5. #5
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    Re: prove that matrix is invertible

    you don't need to use determinants, really.

    suppose that A is singular. this means there is some non-zero vector v, with Av = 0.

    hence Anv = An-1(Av) = An-1(0) = 0.

    so An is singular, as well.
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  6. #6
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    Re: prove that matrix is invertible

    To show that A^n is invertible, you must show that there exist matrix B such that A^nB= BA^n= I where I is the identity matrix.
    Since A is invertible, there exist a matrix C such that AC= CA= I.
    Now, what can you say about A^nC^n= A^{n-1}(AC)C^{n-1} and C^nA^n= C^{n-1}(CA)(A^{n-1})?
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  7. #7
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    Re: prove that matrix is invertible

    thanks, Deveno.

    HallsoiIvy, i need to prove that if p then q, not if q then p.
    i managed to prove the other direction, but thanks anyway.
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