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4x4 matrix A where A^2 and A^3 do not equal A, but A^4 = A

  1. Mar 20, 2013 #1
    The question is, find a 4x4 matrix A where A^2 and A^3 do not equal A, but A^4 = A.

    So far, I have tried matrixes all involving 1's and 0's and -1's. I have also tried to use examples of nilpotent matrixes. However, I have not found anything close.
     
    Last edited by a moderator: Mar 21, 2013
  2. jcsd
  3. Mar 20, 2013 #2

    Dick

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    Think about a matrix that represents a rotation.
     
  4. Mar 20, 2013 #3
    I could come up with a rotation matrix that was 180 degrees but not one that was 360 degrees
     
    Last edited by a moderator: Mar 21, 2013
  5. Mar 20, 2013 #4

    Mark44

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    In future posts, please use the homework template - it's there for a reason. Do not just delete its parts.
     
  6. Mar 20, 2013 #5

    Dick

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    360 degrees is the identity matrix. You don't want that. What angle would work?
     
  7. Mar 20, 2013 #6
    You are right. You'd want 270 degrees right?
     
    Last edited by a moderator: Mar 21, 2013
  8. Mar 20, 2013 #7

    Dick

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    270 degrees is the same as -90. So if A rotates by -90, then A^2 rotates by -180, A^3 by -270 and A^4 by -360, which is the same as the identity again. So A^4 isn't A. It's I. Think about the angle a bit more.
     
  9. Mar 20, 2013 #8
    Oh okay so would it be by 120 degree angles. A2 would be 120 a3 would be 240 A4 would be 360 or the initial A
     
    Last edited by a moderator: Mar 21, 2013
  10. Mar 20, 2013 #9

    Dick

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    Right answer, wrong numbers. A is 120, A^2 is 240, etc. And 360 isn't the initial A. The initial A is 120.
     
  11. Mar 20, 2013 #10
    Okay so now I have the angles, how can I visualize where I want the numbers in the4 x 4 matrix
     
    Last edited by a moderator: Mar 21, 2013
  12. Mar 20, 2013 #11

    Dick

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    There are a lot of places to put them depending on how you want to do the rotation. Do you know how to write a 2x2 matrix that represents a rotation by 120 degrees in the x,y plane? If so, then just write a 4x4 matrix that rotates the first two coordinates by 120 degrees and leaves the other two coordinates fixed.
     
  13. Mar 21, 2013 #12

    Ray Vickson

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    All I see of your response is "222222222". Is that truly your response, or is some kind of hardware/software glitch messing up your answer on my display?
     
  14. Mar 21, 2013 #13

    LCKurtz

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    He fell into a 22222... rotational loop.
     
  15. Mar 21, 2013 #14

    jtbell

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    It's apparently his way of deleting all his posts in the thread. Fortunately you guys quoted enough of what he wrote so the thread still sort of makes sense!
     
  16. Mar 21, 2013 #15

    LCKurtz

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    So maybe it was on a take-home test or for extra credit or something like that and he is trying to cover his tracks? Bring out the horse whips....
     
  17. Mar 21, 2013 #16

    Curious3141

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    His posts have certainly become full of "Number 2". :rofl:
     
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