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Show that Matrix Multiplication is Associative

  1. Jun 28, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that (AB)C=A(BC)

    I am just trying to do this to try to gain some experience with problems like this. I saw in my text that they did a similar example for distributivity using the definition of matrix multiplication, so I thought I could use that approach.




    3. The attempt at a solution

    Let the (i,j)-entry of A be given by aij
    Let the (i,j)-entry of B be given by bij
    Let the (i,j)-entry of C be given by cij

    Then the (i,j)-entry of (AB) is given by

    [tex]\sum_{k=1}^na_{ik}b_{kj}[/tex]

    Here is where I get lost. I was thinking of then writing that the (i,j)-entry of (AB)C would be given by

    [tex]\sum_{k=1}^n(a_{ik}b_{kj})c_{kj}[/tex]


    but I don't think that this works.....and I am not sure why or why not:redface:

    Any hints?

    Thanks!
     
  2. jcsd
  3. Jun 28, 2009 #2

    Office_Shredder

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    You have the right idea, but your element of (AB)C is incorrect. Each entry in AB is going to be a summation, and then each entry in (AB)C should be a summation in which each term contains a summation (specifically, the piece from AB is going to be a summation). So your final equation for the (i,j)th entry of (AB)C should contain a double summation.

    Just walk through the definition of matrix multiplication real carefully and you should be able to get it
     
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