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Undetermined system with no solutions?

  1. Dec 15, 2012 #1
    So I have the system { 2x+3y+z-3v=2 ;x-y+2z+v=0 ;3x+2y+3z-2v=-2} . I have to show that the system has no solutions.. I notice that it has more unknowns than equations so it is an undetermined system.If I form the matrix ? |(2 1 3 ) (3 -1 2) (1 2 3)| I notice that the determinant is diff from zero so this three vectors are linearly indipendent. Now how do I form a matrix using ( 2 0 -2) and these three vectors with the condition that they have a determinant diff from zero? Because if they have a determinant diff from zero,the system has no solutions.
     
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  3. Dec 15, 2012 #2

    haruspex

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    Right, but what are you deducing from that?
    If there are no solutions there must be a linear combination of the equations which eliminates all variables but not all constants. It's fairly easy to spot.
     
  4. Dec 15, 2012 #3
    I wrote the question in the wrong way..the answer on my textbook is " The system has no solutions"..I just have to show if the system has or not solutions...
     
  5. Dec 15, 2012 #4
    But if the system has no solutions the vectors forming the matrix including 2 0 -2 should be linearly indipendent /....
     
  6. Dec 15, 2012 #5

    haruspex

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    Not sure what you mean by 'including 2 0 -2'. Do you mean the 5x3 matrix formed by moving the constants across to the left? The non-existence of solutions does not require all the equations to be linearly independent. Trivial example:
    x - 1 = 0
    x - 2 = 0
    x - 2 = 0
    In the OP, I can see a linear combination of the equations which eliminates all the variables but leaves a contradictory statement regarding the constants. That implies there are no solutions.
    The det of |(2 1 3 ) (3 -1 2) (1 2 3)| is 0.
     
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