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Math Help - Linear algebra proof.

  1. #1
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    Linear algebra proof.

    Question:
    Let A = [aij] be the n x n matrix defined by a_{ii} = k and A_{ij} = 0 if i \neq j. Show that if B is any n x n matrix, then AB = kB..

    My work:
    A = [aij] is not zero only when i = j, as AB = kB would become 0(B) = kB => 0 = 0. Therefore, A = [a_{ii}] since  j = i and a_{ii} = k. Hence,  AB => a_{ii}B => kB.

    What step am I missing? Although it is a relatively simple proof, I am still quite new at proof-writing. Thanks in advance.
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  2. #2
    MHF Contributor

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    Re: Linear algebra proof.

    The " a_{ij}B" is meaningless- it is certainly not what you mean. Also that "=>" indicates that something leads from one to the other. You need to say what that "something" is, not just indicate it.

    You know, of course, that if C= AB then C_{ij}= \sum_{m=1}^n A_{im}B_{mj}. Use the fact that a_{ij}= 0 if i\ne j to reduce that sum to a single term.
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