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Math Help - matrix

  1. #1
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    Question matrix

    Let A and B be n x n matrices such that AB=Isubn . Directly from the definition of linear independence, prove that the colums of B must be linearly independent.

    Your answer should not involve row operations and any theorems on rank/invertibility.
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  2. #2
    MHF Contributor
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    AB = I_n means that A is invertible and B is its inverse. Hence \det{A} \ne 0 and \det{B} \ne 0. Does that help at all?
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  3. #3
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    sorry, i have not learnt det yet. Do you have the other way to prove this? thanks
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  4. #4
    MHF Contributor
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    This might get you thinking along the right lines. The idea is to suppose that the columns of B are not linearly independent and arrive at a contradiction.

    Take A =
    [a b c]
    [d e f]
    [g h i]

    and B =
    [1 2 0]
    [2 4 1]
    [3 6 0].

    Then

    AB =
    [a + 2b + 3c; 2a + 4b + 6c; b]
    [d + 2e + 3f; 2d + 4e + 6f; e]
    [g + 2h + 3i; 2g + 4h + 6i; h]

    But this is impossible because a + 2b + 3c = 1 and 2a + 4b + 6c = 0, but 2a + 4b + 6c = 0 implies a + 2b + 3c = 0. How can you generalize this argument?
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