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Math Help - Non-trivial solution

  1. #1
    Junior Member
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    Non-trivial solution

    x+2y+2z=pz
    2x+y+2z=py
    2x+2y+z=pz

    For what values of p, these equations have non-trivial solution.

    How to solve this question. And also tell me what is meant by non-trivial solution
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  2. #2
    MHF Contributor

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    These can all be written in the form
    x+ 2y+ (2- p)z= 0
    2x+ (1- p)y+ 2z= 0
    2x+ 2y+ (1- p)z= 0
    which obviously has the solution x= y= z= 0. That is the "trivial" solution. If the coefficient matrix \begin{bmatrix}1 & 2 & 2- p \\ 2 & 1- p & 2 \\ 2 & 2 & 1- p\end{bmatrix}
    has determinant 0 (is not invertible) then there will also exist other "non-trivial" solutions.
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