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Eigenvalues and eigenvectors, 3x3 matrix using Remainder and Factor Theorem

  1. Oct 5, 2011 #1
    1. The problem statement, all variables and given/known data

    i = the 3x3 matrix below

    2-λ 0 1
    -1 4-λ -1
    -1 2 0-λ

    Using remainder and factor theorem find the 3 values of λ.

    2. Relevant equations

    |i| = a1|b2c3-c2b3|-a2|a2c3-c2a3|+a3|a2b3-b2a3|

    |a|=ad-bc

    3. The attempt at a solution


    (2-λ) |(4-λ x 0-λ)-(-1x2)|+1|(-1x2)-(4-λ x -1) **because b1 is 0 I've left it out**

    (2-λ)[(4-λ)(0-λ)+2] +1 [-2+(4-λ)]

    (2-λ)(4-λ)(0-λ)+1(4-λ)

    Factorise out (4-λ):

    (4-λ)[(2-λ)(0-λ)+1]

    Multiply [] brackets out (FOIL):

    (4-λ)(λ2-2λ+1)

    ....and now I'm stuck which probably means I've gona wrong somewhere ;-(

    *should I have taken out 0-λ?

    *on the 2nd line of my attempted answer I figured the +2 in the first square brackets cancelled out the -2 in the second square brackets, was I wrong?


    Thanks for any hints and help :)
     
  2. jcsd
  3. Oct 5, 2011 #2

    SammyS

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    2-2λ+1) can be factored.

    If you can't do that, then find the roots: λ2-2λ+1 = 0
     
  4. Oct 6, 2011 #3
    (4-λ) λ(-2λ)

    λ=4 λ=2 λ=0

    Is that wrong?

    I'm not sure how to find the roots of λ2-2λ+1=0 ?
     
  5. Oct 6, 2011 #4

    SammyS

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    Yes, it's wrong
    It's a quadratic equation. Solve it.
     
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