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Math Help - matrix question

  1. #1
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    matrix question

    Let A \in \mathbb{Z}^{m \times m} with det (A) \not =0. Then there exists unique matrix B \in \mathbb{Q}^{m \times m} such that AB=BA=det(A)I and B has integer entries.

    Help me to prove this.
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  2. #2
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    Quote Originally Posted by dimuk View Post
    Let A \in \mathbb{Z}^{m \times m} with det (A) \not =0. Then there exists unique matrix B \in \mathbb{Q}^{m \times m} such that AB=BA=det(A)I and B has integer entries.

    Help me to prove this.
    the matrix B is the adjugate of A. the proof of \text{adj}(A) A=A \ \text{adj}(A)=\det(A)I can be found in any linear algebra textbook. the only thing that i have to add here is that if A \in \mathbb{Z}^{m \times m},

    then from the definition of the adjugate matrix, it's clear that \text{adj}(A) \in \mathbb{Z}^{m \times m} as well. so i don't know why your problem says \text{adj}(A) \in \mathbb{Q}^{m \times m}? we also don't need to have \det(A) \neq 0.


    Edit: just realized that we want a unique B. in this case, the condition \det(A) \neq 0 will take care of that.
    Last edited by NonCommAlg; December 6th 2008 at 03:53 AM.
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  3. #3
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    matrix question

    I think that is for uniqueness of B
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