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Homework Help: Proving a Linear System

  1. Apr 14, 2010 #1
    1. The problem statement, all variables and given/known data

    let x2 + x + 1 be the characteristic polynomial of matrix B

    find B5 using Cayley-Hamilton's Theorem

    2. Relevant equations



    3. The attempt at a solution

    from what i have learn, cayley hamilton theorem is something like this

    B2 + B + I =0

    B(B+I)=I

    so, B-1 = (B+1)

    how can i apply this to make B5? help me please owho
     
  2. jcsd
  3. Apr 14, 2010 #2
    hmm, is my work here valid?

    B-1B = (B+1)B = I

    IB5 = (B+1)B5
     
  4. Apr 14, 2010 #3
    As far as I'm concerned, you mustn't use B^(-1) because you don't know whether B i reversible. Quite likely it is, but I don't know if the fact that the characteristic polynomial has no solutions is enough.

    I have never done such exercise before, but let me try:

    B^2+B+I=0

    B^2=-I-B /*B
    B^3=-B-B^2=-B+I+B=I
    B^4=B^3*B=B
    B^5=B^2=-I-B

    but again, completely not sure :/
     
  5. Apr 14, 2010 #4
    yeaaaa, i dont know B is invertible or not... Silly me.. ahaha

    i guess that is the answer
     
  6. Apr 14, 2010 #5

    Mark44

    Staff: Mentor

    The second equation doesn't follow from the first. The first is equivalent to B2 + B = -I, so B(B + I) = -I
     
  7. Apr 14, 2010 #6
    You have this from Cayley-Hamilton:

    B^2 + B + I = 0
    B^2 = -(B + I)

    Then:

    B^3 = B^2(B) = ..., substitute -(B+I) for B^2, etc, and so on through B^5.
     
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