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Math Help - I don't understand this invertible problem?

  1. #1
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    I don't understand this invertible problem?

    Hello all, this was on my practice final exam for my linear algebra class.

    Let A and B be two nxn square matricies such that AB is invertible. Prove that homogeneous equation (A^2)x=0 Hint:show that A must be invertible,

    The only thing I know is that since A is invertible, by the Inverible MAtrix Theorem, Ax=0 has only trivial solution.

    Help!!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: I don't understand this invertible problem?

    Quote Originally Posted by elpermic View Post
    Hello all, this was on my practice final exam for my linear algebra class.

    Let A and B be two nxn square matricies such that AB is invertible. Prove that homogeneous equation (A^2)x=0 Hint:show that A must be invertible,

    The only thing I know is that since A is invertible, by the Inverible MAtrix Theorem, Ax=0 has only trivial solution.

    Help!!
    If A is invertible, then A^2 is invertible and so A^2x=0 has a unique solution, in fact, x=0. Thus, all you have to show is that A is invertible. But, note that since AB is invertible \det(AB)\ne0. But what do you know about the determinant of a product?
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  3. #3
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    Re: I don't understand this invertible problem?

    This question was before we had learned about determinants. I don't think the professor will allow us to use the determinant to prove it.

    That being said the determinant of a product, for example:

    det(AB) will be equal to det(A) x det(B)
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  4. #4
    Super Member redsoxfan325's Avatar
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    Re: I don't understand this invertible problem?

    Quote Originally Posted by elpermic View Post
    This question was before we had learned about determinants. I don't think the professor will allow us to use the determinant to prove it.

    That being said the determinant of a product, for example:

    det(AB) will be equal to det(A) x det(B)
    So if \det(AB)\neq0, what can you say about \det A and \det B?
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  5. #5
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    Re: I don't understand this invertible problem?

    since detA and detB aren't 0, then A is invertible. And A^k is invertible for any k
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  6. #6
    Super Member redsoxfan325's Avatar
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    Re: I don't understand this invertible problem?

    Bingo
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  7. #7
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    Re: I don't understand this invertible problem?

    Alright thank you very much!

    I just hope that I can do well on my final :/
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  8. #8
    Super Member redsoxfan325's Avatar
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    Re: I don't understand this invertible problem?

    Good luck!
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