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?

det(A)=∑pσ(p)a1p1a2p2...anpn.

I have found this one useful:

Thomas Muir,

Contributions to the History of Determinants 1900-1920.

**Answer**

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.

**Attribution***Source : Link , Question Author : Pekov , Answer Author : hmakholm left over Monica*