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.