Alternating means antisymmetric
First, a key lemma. If det is multilinear and alternating, then swapping two columns flips the sign. Proof: consider det(.., u+v, .., u+v, ..) which is 0 (repeated column). Expand by multilinearity into four terms; the two with a repeated column vanish, leaving det(.., u, .., v, ..) + det(.., v, .., u, ..) = 0. Hence swapping negates. So alternating ⇒ antisymmetric.
This single sign-flip is the engine of everything. Any reordering of the columns is a sequence of swaps, and each swap multiplies det by -1. The net factor depends only on the parity of the permutation — this is precisely the sign of a permutation, sgn(sigma), equal to +1 for an even number of swaps and -1 for an odd number.
Expanding the columns over the basis
Now derive the explicit formula. Write each column in the standard basis: c_j = sum_i a_ij e_i. Plug all n columns into det and use multilinearity to pull every sum out front. We get a giant sum over all choices of one basis index per column. Any choice that repeats a basis vector e_i contributes 0 (alternating). The survivors are exactly the choices that hit each e_i once — i.e. the permutations.
Each surviving term is a product of entries a_{sigma(j), j} times det(e_{sigma(1)}, ..., e_{sigma(n)}). That last det is sgn(sigma) by the swap rule and det(I) = 1. Collecting everything gives the Leibniz formula: det(A) = sum over sigma of sgn(sigma) * product_j a_{sigma(j), j}. Notice we used all three rules exactly once — multilinearity to expand, alternating to kill repeats, normalization to evaluate det(I).
n = 2: permutations of (1,2) are id and the swap.
det = sgn(id)*a11*a22 + sgn(swap)*a21*a12
= (+1)*a11*a22 + (-1)*a21*a12
= a11*a22 - a12*a21 (the a*d - b*c you know)
n = 3: 6 permutations, 3 with sgn +1, 3 with sgn -1:
det = a11 a22 a33 + a12 a23 a31 + a13 a21 a32
- a13 a22 a31 - a11 a23 a32 - a12 a21 a33
General: det(A) = sum_{sigma in S_n} sgn(sigma) * prod_j a_{sigma(j), j}Uniqueness, and why it matters
The derivation never made a choice: starting only from the three rules we were FORCED to the Leibniz sum. So at most one function satisfies all three — that is the uniqueness of the determinant. And the Leibniz formula plainly does satisfy them, so existence holds too. Conclusion: there is exactly one normalized alternating multilinear form on n columns, and we call it det.
Uniqueness is a proof technique, not just a fact. To show two complicated expressions both equal det, you need not match them termwise — just check each is multilinear, alternating, and gives 1 on the identity. We will cash this in repeatedly: it is how we prove det(A^T) = det(A) and the product rule almost without computation.