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Matrix and inverse matrix question?

  1. Feb 7, 2013 #1
    1. The problem statement, all variables and given/known data
    how do I solve this question
    https://www.dropbox.com/s/bn958xnm5483s6u/photo%20%281%29.JPG
    (This link if the image is not visible : https://www.dropbox.com/s/bn958xnm5483s6u/photo (1).JPG)
    just so the equation is not clear it says A2 = λA - 2I

    The inverse should be found through the equation in the question and not through the adjoint method


    2. Relevant equations



    3. The attempt at a solution

    The equation says
    A2 = λA - 2I
    So, I mulitplied by A-1
    This gives
    A2 A-1 = λA A-1 - 2IA-1
    {A-1 . A = I}

    now the equation becomes
    A = λI - 2A-1

    I am stuck here since I cannot simply find A-1 by the adjoint method this equation has two unknowns A-1 and λ . Or maybe I am misinterpreting the question IDK
     
  2. jcsd
  3. Feb 7, 2013 #2

    micromass

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    But [itex]\lambda[/itex] is not an unkown.

    You are given a very specific matrix A. From your specific matrix A, and from your equation [itex]A^2= \lambda A- 2I[/itex], you can deduce what [itex]\lambda[/itex] is.
     
  4. Feb 7, 2013 #3
    How did I miss that . Thanks
     
  5. Feb 7, 2013 #4

    HallsofIvy

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    I see no point in looking for specific values of [itex]\lambda[/itex]. To find the multiplicative inverse of A:
    1) algebraically manipulate the equation, [itex]A^2= \lambda A- 2I[/itex], to get "I" alone on the right.
    2) factor out an "A".
     
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