# Why is the determinant defined in terms of permutations?

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?

$$det(A)=∑pσ(p)a1p1a2p2...anpn. \text{det}(A)=\sum_{p}\sigma(p)a_{1p_1}a_{2p_2}...a_{np_n}.$$

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 $\mathbb R^{n\times n}$ 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.