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For what values of a does the following homogeneous system have a nontrivial solution

  1. Oct 15, 2009 #1
    1. The problem statement, all variables and given/known data

    For what values of a does the system

    (a-1)x + 2y = 0
    2x + (a-1)y = 0

    have a nontrivial solution?

    2. Relevant equations



    3. The attempt at a solution

    Argh... I'm really bad at linear algebra, can't seem to grasp the concepts.

    I know that if I make this into a matrix

    [tex]
    \left[
    \begin{array}{cc}
    a-1 & 2\\
    2 & a-1
    \end{array}
    \right]
    [/tex]

    that the only way the system will have a nontrivial solution is if the reduced row echelon form is not the 2x2 identity, i.e. the matrix is singular. But I don't know how to use those ideas to solve it. Any help would be great!
     
  2. jcsd
  3. Oct 15, 2009 #2

    Pengwuino

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    Gold Member

    Re: For what values of a does the following homogeneous system have a nontrivial solu

    If the determinant is 0, what does that tell you about the system?
     
  4. Oct 15, 2009 #3
    Re: For what values of a does the following homogeneous system have a nontrivial solu

    For some reason, my prof hasn't taught us anything about the determinant. I've read about it, and I sort of have an idea of how it works, but I don't think our prof wants us to use it to solve problems.
     
  5. Oct 15, 2009 #4

    Dick

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    Science Advisor
    Homework Helper

    Re: For what values of a does the following homogeneous system have a nontrivial solu

    Ok, so start reducing it to echelon form. Divide the first row by a-1. Under what conditions can you fail to get ones on the diagonal?
     
  6. Oct 15, 2009 #5
    Re: For what values of a does the following homogeneous system have a nontrivial solu

    my intuition is telling me that a = 3 ... i did it like this:

    [tex]
    \left[
    \begin{array}{cc}
    2 & a-1\\
    a-1 & 2
    \end{array}
    \right]
    \\
    \text{then}
    \\
    \left[
    \begin{array}{cc}
    1 & \frac{a-1}{2}\\
    \frac{a-1}{2} & 1
    \end{array}
    \right]
    [/tex]

    so from this we see that [tex]\frac{a-1}{2} = 1[/tex] because that would make row 2 a multiple of row 1, and subtracting it out would leave a row of zeros. so a = 3.

    does it make sense? i feel like i'm missing something.
     
  7. Oct 15, 2009 #6

    Dick

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    Science Advisor
    Homework Helper

    Re: For what values of a does the following homogeneous system have a nontrivial solu

    That makes a lot of sense. Sure a=3 is a problem. But keep going with the row reduction. Multiply the first row by (a-1)/2 and subtract it from the second row. There is another value of a that creates a problem.
     
    Last edited: Oct 15, 2009
  8. Oct 15, 2009 #7
    Re: For what values of a does the following homogeneous system have a nontrivial solu

    i was thinking that maybe a = 1 is also a problem, but if you substitute a = 1 into the original matrix, you can transform it into the 2x2 identity.

    but if i try to put it into rref i end up having to divide by a-1, and if a=1 then it is undefined.

    so i'm confused about that...
     
  9. Oct 15, 2009 #8

    Dick

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    Science Advisor
    Homework Helper

    Re: For what values of a does the following homogeneous system have a nontrivial solu

    Like I edited my last post to say, just multiply the first row by (a-1)/2 and subtract from the second row. Then ask yourself when you can get a second row of zeros.
     
  10. Oct 15, 2009 #9
    Re: For what values of a does the following homogeneous system have a nontrivial solu

    ah, i see

    a = -1

    thanks again dick. you're really saving my skin.
     
  11. Oct 15, 2009 #10

    Dick

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    Science Advisor
    Homework Helper

    Re: For what values of a does the following homogeneous system have a nontrivial solu

    Well, you are helping by using the hints and thinking about them. Not everyone does that.
     
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