This question is from Artin Algebra, second edition.
After defining determinant and proving many of its properties, author comments about possible alternative approach of defining determinant on P23:
Note: It is natural idea to try defining determinants using compatibility with multiplication and corollary 1.4.13. Since we can write an invertible matrix as a product of elementary matrices, these properties determine the determinants of every invertible matrix. But there are many ways to write a given matrix as such a product. Without going through some steps as we have, it won't be clear that two such products will give the same answer. It isn't easy to make this idea work.
Here, corollary 1.4.13 is this:
But, I can't find the significance of the above note: I have this reasoning: If $A=E_1\cdots E_n=E'_1\cdots E'_k$, but if $\delta(E_1\cdots E_n)=\delta(E_1)\cdots \delta(E_n)$ did not equal to $\delta(E'_1\cdots E'_k)=\delta(E'_1)\cdots \delta(E'_k)$, this would mean that $\delta(A)$ has two different values, but $\delta$ being function, this is absurd.
(I know that the information I provided for this question is not enough, but things are scattered from P20 to P23 of the book. Hence I urge you to refer Artin's Algebra if you have it. I have been wondering about this note for about a year.)