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Math Help - Gaussian elimination (having trouble getting matrix into row echelon form)

  1. #1
    DrZ
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    Gaussian elimination (having trouble getting matrix into row echelon form)

    I am having a lot fo trouble with this question

    Use gaussian elimination to obtain an equivalent system whose coefficient matrix s in row echelon form. If system is consistent and there are free variables, transform it to reduce row echelon form and find al solutions.

    x1 + 3x2 + x3 + x4 = 3
    2x1 - 2x2 + x3 + 2x4 = 8
    x1 - 3x2 + x4 = 5

    I reduced it as far as

    1 -5 0 1 5
    0 -8 -1 0 2
    0 0 0 0 0

    But I cant seem to get it into row echelon form. I tried dividing the 2nd row in half and subtracting (-1) of the top row.

    but then I get this

    1 -5 0 1 5
    -1 1 -1/2 -1 -3
    0 0 0 0 0

    This is my first week taking linear algebra so I guess its expected to run into trouble like this.
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  2. #2
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    Perform the row operation

    R1'=R1-\frac{5}{8}R2.
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  3. #3
    DrZ
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    thanks, I need to start trying fractions in these problems.

    once I do this and reduce it further I get


    1 0 5/8 1 15/4
    0 1 1/8 0 -1/4

    so I am assuming I start solving by doing this

    x2 + 1/8x3 = -1/4 --> x2 = -1/4 -1/8x3

    but now I'm stuck. I cant plug x2 into the first row since there is 0 x2. Did I start this incorrectly?
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  4. #4
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    x3 is a "free variable" in this system. x1 and x2 are "dependent" variables. this is to say, you have to pick a value for x3 before you know what x1 and x2 have to be.

    your answer should be of the form (a+bt, c+dt, t), the solution space is 1 dimensional (a line in 3-space), rather than 0-dimensional (a single point).
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  5. #5
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    Look at some examples in your linear algebra book to see how they handle free variables.
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