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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- May 21st 2011, 04:28 AMStuck ManGaussian 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?

- May 21st 2011, 05:08 AMTinyboss
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.

- May 21st 2011, 05:24 AMStuck Man
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.

- May 21st 2011, 06:12 AMStuck Man
Have you misunderstood my question? Is the bottom right entry of a matrix in row echelon form the determinant of the original matrix?

- May 21st 2011, 07:20 AMtopsquark
- May 21st 2011, 07:25 AMStuck Man
No but is it the determinant of the original matrix before it was converted?

- May 21st 2011, 08:02 AMtopsquark
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:

$\displaystyle \begin{pmatrix} 2 & -1 \\ 3 & 2 \end{pmatrix}$

-Dan - May 21st 2011, 08:27 AMStuck Man
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.

- May 21st 2011, 09:00 AMStuck Man
http://img8.imageshack.us/img8/4772/capture1eq.jpg

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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? - May 21st 2011, 09:03 AMStuck Man
http://img819.imageshack.us/img819/3217/capture1rr.jpg

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I do not understand the logic to the final part of the answer. - May 21st 2011, 09:16 AMtopsquark