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Show that if A is an n x n matrix

  1. Dec 11, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that if A is an n x n matrix whose kth row is the same as the kth row of In, then 1 is an eigenvalue of A.


    2. Relevant equations None that I know of.



    3. The attempt at a solution I tried creating an arbitrary matrix A and set it up to to find the det(A-λI) but that got me no where.
     
  2. jcsd
  3. Dec 11, 2012 #2

    mfb

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    Re: Eigenvalues

    If you know that A and At have the same eigenvalues, you can work with the original definition of eigenvalue: There is a vector x such that Atx = 1x.
     
  4. Dec 11, 2012 #3
    Re: Eigenvalues

    I'm still not understanding what to do with that...
     
  5. Dec 11, 2012 #4

    Dick

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    Re: Eigenvalues

    Think about an expansion by minors along the kth row of A-λI. You want to show 1-λ is a factor of the characteristic polynomial.
     
  6. Dec 11, 2012 #5

    mfb

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    Re: Eigenvalues

    It is easy to find a specific vector x with Atx = 1x. That shows that x is an eigenvector of At with eigenvalue 1 => At has 1 as eigenvalue => A has 1 as eigenvalue.
     
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