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Math Help - Gaussian elimination and determinant

  1. #1
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    Gaussian elimination and determinant

    When I do gaussian elimination on a 3x3 matrix to get a matrix in row echelon form is the bottom right entry the determinant of the original matrix?
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    Senior Member Tinyboss's Avatar
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    No. Take a matrix that's already in row-echelon form. Since it's triangular, its determinant is the product of the diagonal entries. Now do Gaussian elimination to put it into row-echelon form (of course you do nothing in this step, but that doesn't matter). Now if your conjecture was true, the determinant would be equal to the bottom-right entry.
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    What exactly is the product of the diagonal entries? I've looked at a couple of examples I've done and the bottom right entry is the determinant.
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    Have you misunderstood my question? Is the bottom right entry of a matrix in row echelon form the determinant of the original matrix?
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Stuck Man View Post
    Have you misunderstood my question? Is the bottom right entry of a matrix in row echelon form the determinant of the original matrix?
    Consider, as Tinyboss suggested, a matrix in upper triangular form:
    \begin{pmatrix} 1 & 0 & 3 \\ 0 & 4 & -2 \\ 0 & 0 & 18 \end{pmatrix}

    Is the determinant of this matrix 18?

    -Dan
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    No but is it the determinant of the original matrix before it was converted?
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    Quote Originally Posted by Stuck Man View Post
    No but is it the determinant of the original matrix before it was converted?
    Tinyboss' suggestion (which I'm merely giving you a concrete example of) is that the matrix I gave you can be put into echelon form (thought it's already there) and that the determinant is not the lower right element. However I'm presuming you have a reason for saying this. So let's do this. Work out the echelon form of the following matrix and compare the lower right element with the determinant of the original:
    \begin{pmatrix} 2 & -1 \\ 3 & 2 \end{pmatrix}

    -Dan
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    I have found them to be equal but I can see now that it won't always be equal. On the other hand the bottom right entry may be the determinant multiplied by a scalar. This is what I find when there are variables.
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    The entry at the bottom right can be used to answer the second part of this question. I understand how to use the determinant but what is the logic here?
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    I do not understand the logic to the final part of the answer.
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  11. #11
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Stuck Man View Post


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    I do not understand the logic to the final part of the answer.
    If a = 5/2, what is the determinant of the echelon form matrix? In this case we get the same determinant for the original coefficient matrix.

    -Dan
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