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Math Help - Linear algebra proof

  1. #1
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    Linear algebra proof

    Let A be a m * p matrix whose columns all add to the same total s; and B be a p * n matrix whose columns all add to the same total t: Using summation notation prove that the n columns of AB each total st:
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  2. #2
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    Quote Originally Posted by ulysses123 View Post
    Let A be a m * p matrix whose columns all add to the same total s; and B be a p * n matrix whose columns all add to the same total t: Using summation notation prove that the n columns of AB each total st:
    By definition of matrix multiplication, the entries of AB are given by \sum_{i=1}^p a_{j,i} b_{i,k}.

    We seek the sum of an arbitrary column of AB, given by \sum_{j=1}^m\left(\sum_{i=1}^p a_{j,i} b_{i,k}\right)

    =\sum_{i=1}^p \left( b_{i,k} \sum_{j=1}^m a_{j,i}\right)

    =\sum_{i=1}^p \left( b_{i,k} s\right)

    =s\sum_{i=1}^p  b_{i,k}

    =st .
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  3. #3
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    Thanks, what i was doing was expressing one column in Matrix A, by the sum of all columns divided by p, which was making it difficult to then simplify the expression for AB.
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