Row echelon form and determinant

Question

I computed row echelon forms(without fractions) of $3\times3$ matrices and i noticed that the value of the last number on the diagonal $(position $(3,3)$ )$ is equal to the determinant.Is this true, why?

Edit: i used Maple for the calculations
This seems to work for all matrices

Edit: @algebraicpavel i computed the REF and smith form of the matrix that you posted this is very weird