Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of permutations?
I have found this one useful:
This is only one of many possible definitions of the determinant.
A more “immediately meaningful” definition could be, for example, to define the determinant as the unique function on Rn×n such that
- The identity matrix has determinant 1.
- Every singular matrix has determinant 0.
- The determinant is linear in each column of the matrix separately.
(Or the same thing with rows instead of columns).
While this seems to connect to high-level properties of the determinant in a cleaner way, it is only half a definition because it requires you to prove that a function with these properties exists in the first place and is unique.
It is technically cleaner to choose the permutation-based definition because it is obvious that it defines something, and then afterwards prove that the thing it defines has all of the high-level properties we’re really after.
The permutation-based definition is also very easy to generalize to settings where the matrix entries are not real numbers (e.g. matrices over a general commutative ring) — in contrast, the characterization above does not generalize easily without a close study of whether our existence and uniqueness proofs will still work with a new scalar ring.