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Characteristic polynomial

  1. Jan 5, 2007 #1
    And again a question:

    L is a field for which [tex] a \in L [/tex]. The matrix

    [tex]
    A = \frac{1}{2}\left( {\begin{array}{*{20}c}
    1 & 1 & 1 & 1 \\
    1 & a & { - 1} & { - a} \\
    1 & { - 1} & 1 & { - 1} \\
    1 & { - a} & { - 1} & a \\
    \end{array}} \right)
    [/tex]

    has the characteristic polynomial

    [tex]
    x^4 - \left( {a + 1} \right)x^3 + \left( {a - 1} \right)x^2 + \left( {a + 1} \right)x - a
    [/tex]

    I need to show that this information is correct for a=1 in any field.

    My problem is that when i calculate the polynomial for a=-1 I end up with another characteristic polynomial than the one given. Maybe i'm going about it the wrong way. Suggestions or pointers are very welcome
     
  2. jcsd
  3. Jan 6, 2007 #2

    Hurkyl

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    I don't understand the question...

    And if you want to show something is correct for a=1, then why are you looking at a=-1?
     
  4. Jan 6, 2007 #3

    mathwonk

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    i conjecture he meant the characteristic polynomial is accurate for a any element of any field, and yet it failed for a = -1.
     
  5. Jan 7, 2007 #4
    Oops, should have stated:

    Show that this is correct for a=-1 in any field.
     
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