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Algebraic/Matrix Manipulation (linear algebra)

  1. Jan 5, 2013 #1
    1. The problem statement, all variables and given/known data

    I have attached the relevant question as an image (for sake of ease)

    2. Relevant equations



    3. The attempt at a solution

    Also attached, in blue.


    Thanks a lot for any help at all!
     

    Attached Files:

  2. jcsd
  3. Jan 5, 2013 #2

    mfb

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    Staff: Mentor

    You could consider ##B^k Y = 0##. Which property does B need to get non-trivial solutions for Y? How is that related to a?
    ##Y=(B-I_2)X##. How do you get Y=0 (which is a solution to the equation above) with non-trivial X? How is that related to a?
     
  4. Jan 6, 2013 #3
    Perhaps the very beginning is where I'm having trouble, I don't know what times a matrix would be equal to zero. Is it the transpose? Or perhaps a matrix whose determinant is zero?
     
  5. Jan 6, 2013 #4

    HallsofIvy

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    If you have studied matrices at all then you should know this basic property: the equation Ax= y has a unique solution if and only if A is invertible: [itex]x= A^{-1}y[/itex]. And that is only true if A has non-zero determinant.

    The equation Ax= 0 always has the "trivial" solution, x= 0. It has other solutions if and only if its determinant is 0.
     
  6. Jan 6, 2013 #5
    You're right, I should know that basic property :) I've been cramming too much this semester so I tend to forget.

    Thanks a lot it makes good sense.
     
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