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Row Reduction over indicated field

  1. Feb 3, 2012 #1

    I am taking linear algebra and the teacher threw some problems at us that I cannot seem to figure out or find examples for online. Also, the textbook doesn't even cover the material.

    The problem states:
    Using Gauss-Jordan elimination, solve the following system with coefficients in indicated field.

    In Z3:

    2x + y = 1
    x + y = 2

    Now I understand completely how to row reduce this system, but am not sure how to proceed in Z3.

    Any help appreciated. Thanks!
  2. jcsd
  3. Feb 3, 2012 #2
    The best thing to do here is to solve the system as you normally would and then interpret the solution in [itex]\mathbb{Z}_3[/itex].

    So, how would you proceed normally?? Just write the first step, don't solve it completely yet.
  4. Feb 3, 2012 #3
    Thanks for the reply.

    The first thing I would do is swap row 1 and row 2 to get a leading 1 in row 1.

    [1 1 l 2]
    [2 1 l 1]

    From here would take R2-2R1.
  5. Feb 3, 2012 #4
    OK, so do R2-2R1. But be aware that the arithmetic is in [itex]\mathbb{Z}_3[/itex].

    So, for example, the second column gives us 1-2*1=-1=2.
  6. Feb 3, 2012 #5


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    or, (to keep it straight as you go) realize that -2 in Z3 is actually 1, so instead of R2-2R1, use R2+R1 (which gives you 3 = 0 in the first column).

    that is, instead of aiming to "zero out" entries, you just want to get them to the nearest multiple of 3 (which amounts to the same thing in Z3). with a little practice, you can see how to do this and always keep all the entries positive, which might help a bit.
  7. Feb 20, 2012 #6
    Thanks for the reply. The part I don't understand is how -1 = 2 and -2 = 1 in Z3. Would you be able to explain this part to me?
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