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Math Help - Linear Algebra Proof - Prove that when A+D=0, D is unique.

  1. #1
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    Linear Algebra Proof - Prove that when A+D=0, D is unique.

    I need to prove that for each m \times n matrix A, there exists a unique m \times n matrix D such that A+D=O. Where O is the m \times n zero matrix.

    I have a textbook that states the proof simply as:

     \textrm{For } A=[a_{ij}] \textrm{ Let } D=[-a_{ij}].

    I'm not sure how to construct a a better proof for this result. Could anyone offer a starting point?
    Last edited by Fourier; September 5th 2007 at 03:59 PM.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let A=(a_{ij}), \ D=(x_{ij})
    Then A+D=(a_{ij}+x_{ij}).
    A+D=O_{mn}\Rightarrow a_{ij}+x_{ij}=0\Rightarrow x_{ij}=-a_{ij}\Rightarrow D=(-a_{ij})

    Suppose D' is another matrix such as A+D'=D'+A=O_{mn}.
    Then D'=D'+O_{mn}=D'+(A+D)=(D'+A)+D=O_{mn}+D=D
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  3. #3
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    Thanks for giving the alternate proof.
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