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Show that matrix A is not invertible by finding non trivial solutions

  1. Feb 13, 2013 #1
    1. The problem statement, all variables and given/known data

    The 3x3 matrix A is given as the sum of two other 3x3 matrices B and C satisfying:1) all rows of B are the same vector u and 2) all columns of C are the same vector v.

    Show that A is not invertible. One possible approach is to explain why there is a nonzero vector x satisfying both Bx = 0 and Cx = 0.

    ^^I'm having a hard time seeing why Bx=0 and Cx=0 should have nonzero solutions. I envision a matrix {{u1,u2,u3},{u1,u2,u3},{u1,u2,u3}} * some column vector = 0 but I'm just not seeing how to go about this when u1,u2,u3 could be anything.


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 13, 2013 #2
    Re: Show that matrix A is not invertible by finding non trivial soluti

    Bx=0 and Cx=0 are homogeneous system of equations. So there are only two posibilities for their solutions. 1. Either the solutions are trivial and unique or 2. Infinitely many solutions.

    Ask yourself if B is invertible? If it is, then multiply both sides of Bx=0 by its inverse to obtain only the trivial solution exists. If A is not invertible, then it must case 2.

    Let me know if it helps.
     
  4. Feb 13, 2013 #3
    Re: Show that matrix A is not invertible by finding non trivial soluti

    hmm, you've given me something to think about. So I can see how B is not invertible because well Gaussian elimination on it fails, but what about C? I don't think you can immediately see it by trying to do the more familiar elimination steps, so is it enough to say that in a way it a dependent matrix?
     
  5. Feb 13, 2013 #4

    vela

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    Re: Show that matrix A is not invertible by finding non trivial soluti

    Try writing out Cx=0 explicitly.
     
  6. Feb 13, 2013 #5
    Re: Show that matrix A is not invertible by finding non trivial soluti

    I am not sure what you mean by elemination failing. However, elimination give many rows of zero and hence infinitely many solutions for Bx=0. Consider doing elimination on C for a 2x2 matrix with the same columns and see what are your solutions.

    If you have learn the determinant, then the determinant also gives you the answer imeddiately.
     
  7. Feb 13, 2013 #6
    Re: Show that matrix A is not invertible by finding non trivial soluti

    First of all, what do you know about a matrix that has linearly dependent row or column vectors?
     
  8. Feb 13, 2013 #7
    Re: Show that matrix A is not invertible by finding non trivial soluti

    oh I see, if you try to eliminate one you end up eliminating all the others in the row as well
     
  9. Feb 13, 2013 #8
    Re: Show that matrix A is not invertible by finding non trivial soluti

    Sorry, I should have been more clear. That is what I meant.
     
  10. Feb 13, 2013 #9
    Re: Show that matrix A is not invertible by finding non trivial soluti

    Ok. I think I got it. Thank you all for your quick replies!
     
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