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Linear algebra, eigenvectors and eigenvalues

  1. Apr 14, 2010 #1
    If v is an eigenvector of an invertible matrix A, which of the following is/are necessarily true?

    (1) v is also an eigenvector of 2A
    (2) v is also an eigenvector of A^2
    (3) v is also an eigenvector of A^-1

    A) 1 only
    B) 2 only
    C) 3 only
    D) 1 and 3 only
    E) 1,2 and 3

    I am pretty sure 2 is true because if we look at Av = (lamba)v.
    A^2v = A(lambda)v = (Av)(lambda) = (lambda)v(lambda) = (lambda)^2 v

    So A^2v = (lambda)^2v. So that should be that v is an eigenvector for A^2 as well. I am not sure about the others can someone help?
     
  2. jcsd
  3. Apr 14, 2010 #2

    Dick

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    You did so well on b), isn't a) just as easy? (2A)v=??? For c) use A^(-1)A=I.
     
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