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Math Help - Solutions for a homogeneous system

  1. #1
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    Solutions for a homogeneous system

    I need to determine for what values of "a" the system will have nontrivial solutions.

    I know that nontrivial means that both values of x and y can not be zero.

    I am able to, by trial and error, determine values of "a" which provide nontrivial solutions. But, I am sure that I am suppose to come up with a systematic method to determine this. Since I am in a matrices and power function class, I am sure I am not suppose to just use my current method.

    My system I am currently working on is:

    (1- a)x + 2y = 0

    3x + (2 - a)y = 0

    Any help would be much appreciated. Frostking
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  2. #2
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    Quote Originally Posted by Frostking View Post
    I need to determine for what values of "a" the system will have nontrivial solutions.

    I know that nontrivial means that both values of x and y can not be zero.

    I am able to, by trial and error, determine values of "a" which provide nontrivial solutions. But, I am sure that I am suppose to come up with a systematic method to determine this. Since I am in a matrices and power function class, I am sure I am not suppose to just use my current method.

    My system I am currently working on is:

    (1- a)x + 2y = 0

    3x + (2 - a)y = 0

    Any help would be much appreciated. Frostking
    Writing the matrix equation gives.

    \begin{bmatrix}(1-a) & 2 \\ 3 & (2-a) \end {bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix}0 \\ 0 \end{bmatrix}

    This equation will have a unique solution when the matrix is invertable. A matrix is invertable when its determinant is not zero.

    \begin{vmatrix}(1-a) & 2 \\ 3 & (2-a) \end {vmatrix}=(1-a)(2-a)-6

    a^2-3a-4=0 \iff (a-4)(a+1)=0

    So you will get nontrivia(an infinite number of)l solutions when a=4 or a=-1
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