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Gaussian Elim Problem

  1. Jun 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Examine the solutions of

    [tex]\left[\begin{array}{c}x_1-x_2+x_3=c\\2x_1-3x_2+4x_3=0\\3x_1-4x_2+5x_3=1\end{array}\right][/tex]

    [tex]\text{when }c=1\text{ and }c\ne1[/tex]


    3. The attempt at a solution

    [tex]\left[\begin{array}{cccc}1& -1& 1& c\\2& -3& 4& 0\\3& -4& 5& 1\end{array}\right][/tex]

    [tex]\left[\begin{array}{cccc}1& -1& 1& c\\0& -1& 2& -2c\\0& -1& 2& -2c\end{array}\right][/tex]

    [tex]\left[\begin{array}{cccc}1& -1& 1& c\\0& 1& -2& 2c\\0& 0& 0& 0\end{array}\right][/tex]


    What is the next step here?
     
  2. jcsd
  3. Jun 27, 2009 #2

    Pengwuino

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    Gold Member

    What this step is telling you is that x3 is arbitrary. Continue elimination if you want, but you can construct your solution from this. For example, the 2nd row shows that x2 = 2c +2x3.
     
  4. Jun 27, 2009 #3
    The entry at the far bottom right of the second matrix should be 1 - 3c.
     
  5. Jun 27, 2009 #4

    Pengwuino

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    Gold Member

    oops yah I didn't notice that. it is 1-3c which changes the situation...
     
  6. Jun 27, 2009 #5
    Doh!!!!!

    edit: Hey!! PF changed my DOH from all Caps!
     
  7. Jun 27, 2009 #6
    [tex]
    \left[\begin{array}{cccc}1& -1& 1& c\\0& -1& 2& -2c\\0& -1& 2& 1-2c\end{array}\right]
    [/tex]

    [tex]
    \left[\begin{array}{cccc}1& -1& 1& c\\0& 1& -2& 2c\\0& 0& 0& 1\end{array}\right]
    [/tex]

    Now.... have I made another silly error?

    Is the elimination finished?
     
  8. Jun 27, 2009 #7

    Pengwuino

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    Gold Member

    No, you made another mistake. You're subtracting 3 times the first row so you should have 1-3c as the bottom right coefficient, not 1-2c.
     
  9. Jun 27, 2009 #8
    Ohh jeesh...you both already said that....last stage becomes:

    [tex]

    \left[\begin{array}{cccc}1& -1& 1& c\\0& 1& -2& 2c\\0& 0& 0& (1-c)\end{array}\right]

    [/tex]

    Now the question makes way more sense.

    c=1-->infinite solutions
    c[itex]\ne[/itex]1-->inconsistent eqs

    Thanks people! I should go to bed... but I think I can squeeze in one more stupid question tonight!!
     
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