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?
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.
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
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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.