One intuitive way to approach studying the determinant of a given matrix $M$ is to inspire its formal definition in the signed volume of applying the corresponding transformation to the unit $n$-cube.
By noticing that $M = E_1...E_nM^R$ where $E_i$ is an elementary matrix and $M^R$ is the row reduced echelon form of $M$ one could define $\mathrm{det}(M)$ as $\frac{\prod^{}_{} \mathrm{diag}(M^R)}{p}$, where $p = \prod^{}_{}\mathrm{det}(E_i)$.
The issue is that one could arrive at $M^R$ in more than one way, so that the elementary matrices may differ, so might $p$. How could one then prove that regardless of how one computes $M^R$ the product of the resulting elementary matrices will always be the same, that is, how could one prove that the above definition of the determinant is well defined?
Would appreciate any help.