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Minimal and characteristic polynomial

  1. Feb 15, 2005 #1
    Find the characteristic and minimal polynomials of
    A=[[0,1,1][1,0,1][1,1,0]] (3x3 matrix)

    So when I work out my characteristic polynomial I went
    det(xI-A)= det[[x,1,1][1,x,1][1,1,x]]
    = x(x^2-1)-1(x-1)+1(1-x)
    = x^3-3x+2
    = (x+2)(x-1)^2
    It's odd because I worked this out several times, and by Cayley Hamilton's theorem it says that a characterstic polynomial of a matrix is also an annihilating polynomial for that matrix, and I tried plugging in A to the characteristic polynomial and it didn't give me the 0 matrix.

    My prof's answer for the characteristic polynomial is (t-2)(t+1)^2
    and her minimal polynomail is (t+1)(t-2)

    Which works.

    I'm really confused, can someone please tell me what I did wrong.

    thanks in advance
     
  2. jcsd
  3. Feb 16, 2005 #2

    AKG

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    Note that it's det(xI - A), not det(xI + A), i.e. this line is wrong:

    det(xI-A)= det[[x,1,1][1,x,1][1,1,x]]
     
  4. Feb 16, 2005 #3
    How is this line wrong ??

    A=[[0,1,1][1,0,1][1,1,0]]
    xI=[[x,0,0][0,x,0][0,0,x]]


    so xI-A=[[x-0,1,1][1,x-0,1][1,1,x-0]]
    =[[x,1,1][1,x,1][1,1,x]]


    I'm pretty sure this looks ok

    Thanks for any help in advance
     
    Last edited: Feb 16, 2005
  5. Feb 16, 2005 #4

    HallsofIvy

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    Then look again!!!

    xI- A=[x-0,0-1,0-1][0-1,x-0,0-1][0-1,0-1,x-0]
    =[x, -1, -1][-1, x, -1][-1, -1, x].
     
  6. Feb 16, 2005 #5
    omg lol sorry about that
     
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