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Matrix to an exponent

  1. Oct 9, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider the matrix

    cos(3*pi/17) -sin(3*pi/17)
    S = sin(3*pi/17) cos(3*pi/17)

    Does there exist a positive integer n such that Sn = I where I is the 2x2 identity? If so, what is the smallest such integer? Explain.

    Excuse the poor matrix formatting. I do not know how to use the latex formatting to put it into pretty print.

    2. Relevant equations
    Not sure....


    3. The attempt at a solution

    Where should I start? I really have no idea.
     
  2. jcsd
  3. Oct 9, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi DmytriE! :smile:

    Hint: suppose S is

    Code (Text):
    cosθ -sinθ
    sinθ cosθ
    What is S2 ? S3 ? etc? :wink:
     
  4. Oct 9, 2012 #3
    Let's suppose that n = 10. I don't think I have to multiply S by S 10 times to get the answer. Unfortunately the answer is not 10. How would S2, S3 help me figure it out?

    S2:
    UL:cos2(θ) - sin2(θ)
    UR: -2sin(θ)cos(θ)
    LL: 2sin(θ)cos(θ)
    LR: -sin2(θ)+cos2(θ)

    Each abbreviation represent the place in the matrix that they would appear. UL - Upper left, etc.

    S3:
    Alot of sines and cosines.
     
  5. Oct 9, 2012 #4

    Mark44

    Staff: Mentor

    That's not what tiny-tim is suggesting. Your matrix represents a certain kind of transformation.

    Instead of thinking about what S, S2, S3, etc. are (in terms of their matrix representations), think about what they do to a vector they multiply.
     
  6. Oct 9, 2012 #5
    Thanks for the help! This forum really is the best!
     
  7. Oct 10, 2012 #6

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi DmytriE! :smile:

    (just got up :zzz:)
    have you got it now?

    if not, use standard trigonometric identities :wink:
     
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