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Math Help - [SOLVED] Multiplication of matrices

  1. #1
    MHF Contributor arbolis's Avatar
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    [SOLVED] Multiplication of matrices

    The problem has certainly already be solved but I didn't find it in a short research.
    Anyway I'm just asking for a check-result.
    Prove that if A and B are n\times m matrices and C is a m\times q one, then (A+B)C=AC+BC.
    My attempt : By definition, (A+B)_{ij}=A_{ij}+B_{ij} so (A+B)_{ij}C_{ij}=\sum_{k=1}^n [A_{ik}+B_{ik}]C_{kj} (call this 1) =\sum_{k=1}^n A_{ik}C_{kj}+B_{ik}C_{kj} (call this 2) =AC+BC.
    I'm not sure if I can pass from 1 to 2 as I did or I need more calculus/or explanation.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by arbolis View Post
    The problem has certainly already be solved but I didn't find it in a short research.
    Anyway I'm just asking for a check-result.
    Prove that if A and B are n\times m matrices and C is a m\times q one, then (A+B)C=AC+BC.
    My attempt : By definition, (A+B)_{ij}=A_{ij}+B_{ij} so (A+B)_{ij}C_{ij}=\sum_{k=1}^n [A_{ik}+B_{ik}]C_{kj} (call this 1) =\sum_{k=1}^n A_{ik}C_{kj}+B_{ik}C_{kj} (call this 2) =AC+BC.
    I'm not sure if I can pass from 1 to 2 as I did or I need more calculus/or explanation.
    Are A_{ik},~B_{ik},~C_{kj}\in\mathbb{R}?

    If so, I'm pretty sure it is safe to assume that \sum_{k=1}^n [A_{ik}+B_{ik}]C_{kj}=\sum_{k=1}^n A_{ik}C_{kj}+B_{ik}C_{kj}=AC+BC, since we can apply the distributive property to real numbers.

    --Chris
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  3. #3
    MHF Contributor arbolis's Avatar
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    Are ?
    I think so, or \mathbb{C}. So that means the demonstration is right.
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  4. #4
    Moo
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    A Cute Angle Moo's Avatar
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    Hi guys,

    There is a little problem with that. Just to be picky, but it looks important to me >.<

    Let A=\left[A_{ij}\right]_{\substack{1\le i \le n \\ 1 \le j \le {\color{red}m}}}
    etc...

    It is clear (well in fact you gotta know the conventions) that the second coordinate of A_{ij} goes from 1 and m and not 1 and n.

    So (A+B)_{ij}C_{ij}=\sum_{k=1}^n [A_{ik}+B_{ik}]C_{kj}<br />
is rather (A+B)_{ij}C_{ij}=\sum_{k=1}^{{\color{red}m}} [A_{ik}+B_{ik}]C_{kj}

    The first coordinate = # of the row.
    The second coordinate = # of the column.

    A matrix (n,m) is meant to have n rows and m columns.
    Last edited by Moo; September 15th 2008 at 10:09 AM. Reason: a mistake of copy and paste
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  5. #5
    MHF Contributor arbolis's Avatar
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    Thank you Moo for pointing that out. If I have something similar in a exam I'll take care of not doing this error.
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