1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    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=-I-B /*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


    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)


    B^3 = B^2(B) = ..., substitute -(B+I) for B^2, etc, and so on through B^5.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook