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Patterns in matrices

  1. Oct 29, 2007 #1
    K guys heres the problem

    P= (3 1
    1 3)

    S=(4 2
    2 4)

    Calculate P^n and S^n for other values of n and describe any patterns you see.
    I tried this one for about an hour and got a little bit. I just want to see what you can get out of it. Maybe I missed something. Please Help! thanks
     
    Last edited: Oct 29, 2007
  2. jcsd
  3. Oct 29, 2007 #2

    Zurtex

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    What values of n did you calculate it for? Can you show us a few examples and post anything if you anything, or not if you don't.
     
  4. Oct 29, 2007 #3
    i calculated it out for 1,2,3,4,and 5 its really hard to post on my computer. do you have any ideas for finding a general form? because that is the basis of the problem
     
  5. Oct 29, 2007 #4
    i'll see what i can do about the examples
     
  6. Oct 29, 2007 #5
    P^3= (36 28
    28 36)
    P^4= (136 120
    120 136)
    P^5= (528 496
    469 528)
     
  7. Oct 29, 2007 #6

    Zurtex

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    (3 1)2
    (1 3)
    =
    (10 6)
    (6 10)


    (3 1)3
    (1 3)
    =
    (36 28)
    (28 36)


    (3 1)4
    (1 3)
    =
    (136 120)
    (120 136)


    Do you not spot a pattern?

    Are you familliar with proof by induction?
     
  8. Oct 29, 2007 #7
    S^2= (20 16
    20 16
    S^3= (112 104
    112 104)
    S^4= (656 640
    640 656)
    S^5= (3904 3872
    3872 3904)
     
  9. Oct 29, 2007 #8
    no i'm not sorry i'm trying to learn this. its an assignment my teacher gave us and told us to run with. i saw one pattern but i don't really know how to explain it. i noticed that the first term in each matrix differed from the second term by 2^n. thats all i got by looking at it
     
  10. Oct 29, 2007 #9

    Zurtex

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    That's quite cool, do you know how to summate terms like this:

    [tex]\sum_{x=1}^n x[/tex]
    ?

    (Not this particular example, but that sort of style of summation)
     
  11. Oct 29, 2007 #10
    yes i do
     
  12. Oct 29, 2007 #11
    yes she has taught us that but i don't know what that has to do with it?
     
  13. Oct 29, 2007 #12

    Zurtex

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    Think about trying to multiply the matrix "n times then". Perhaps start with an easy example then like:

    (1 1)n
    (1 1)
    =
    Code (Text):

    (1 1) (1 1) (1 1) ... (1 1)
    (1 1) (1 1) (1 1)     (1 1)
     
    (Try actually writting what's happening in each element, you should get a bit of a long sum, that you can calculate).
     
  14. Oct 29, 2007 #13
    ok i did that but i'm still not getting how to work that with my original problem
     
  15. Oct 30, 2007 #14

    Zurtex

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    Well it's the same princaple, if you get a summation form in each of the element, you've worked out what it is, more over you may be able to put it in a closed form if you understand how to do the summations.
     
  16. Oct 30, 2007 #15
    ok thank you, i'll try that today i'm pretty sure i'll be able to work it out now. That helped alot.
     
  17. Oct 30, 2007 #16
    hey i couldn't find any patterns that way. did you find anything?
     
  18. Oct 31, 2007 #17

    Zurtex

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    Yeah, I worked them both out pretty quickly, just trying to help you along rather than give the answer. I don't know how else to help you without just saying the answer :/
     
  19. Nov 1, 2007 #18
    ok well i turned in the paper today hopefully it is right. the general form i came up with was like a scalar or 2^(n-1) (k^n+1 k^n-1)
    (k^n-1 k^n+1)
     
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