### Theory of Determinant

##### 3. Existence and uniqueness

Based on the three properties in the definition, we derived the formula

detA = ∑all permutations (i1,i2,...,in) of (1,2,...,n) sign(i1, i2, ..., in) ai11ai22...ainn.

We have two remarks.

1. Since the functions satisfying the three properties must be given by the explicit formula, the determinant (if it exists) must be unique;
2. We may prove the existence of det by verifying that the formula indeed satsify the three properties.

In the subsequent verification that the formula satisfies the three properties, the formula will be temporarily denoted D(A) (we are not supposed to use the det until the properties are verified).

Multilinear Since linear combinations of linear functions are linear, we only need to verify that for each fixed permutation (i1, i2, ..., in), the function ai11ai22...ainn is linear in columns of A. For the linearity in [col 1], we fix entries in [col 2], ..., [col n] and consider entries in [col 1] as changing. In our case, ai22...ainn is a fixed number and ai11 changes. Then the composition

 a1 = [ a11 ] → ai11 → ai11ai22...ainn a21 : an1

is linear because the first projection is linear and the second scalar multiplication by a fixed number is also linear. The reason for the linearity in the other columns is similar.

Alternating Let B be obtained from A by exchanging [col 1] and [col 2]. Then ai2 is an entry in [col 1] of B, ai1 is an entry in [col 2] of B, and the others are not changed. The change leads to

D(B) = ∑all permutations (i1,i2,...,in) of (1,2,...,n) sign(i1, i2, ..., in) ai12ai21ai33...ainn
= ∑all permutations (i1,i2,...,in) of (1,2,...,n) sign(i1, i2, ..., in) ai21ai12ai33...ainn.

Now we claim

sign(i2, i1, ..., in) = - sign(i1, i2, ..., in).

To see why, let

(i1, i2, ..., in) → ... → (1, 2, ..., n)

be a sequence of transpositions changing (i1, i2, ..., in) to (1, 2, ..., n). Then by adding the transposition (i2, i1, ..., in) → (i1, i2, ..., in) to the front, we get a sequence

(i2, i1, ..., in) → (i1, i2, ..., in) → ... → (1, 2, ..., n)

of transpositions changing (i2, i1, ..., in) to (1, 2, ..., n). The upshort is that the number of transpositions in the two sequences have different parity. Thus sign(i2, i1, ..., in) and sign(i1, i2, ..., in) have different signs. With the help of the discussion on sign, we conclude (in the second equality, we relabeled i2 as i1 and i1 as i2)

D(B) = ∑all permutations - sign(i2, i1, ..., in) ai21ai12ai33...ainn
= - ∑all permutations sign(i1, i2, ..., in) ai11ai22...ainn
= - D(A).

Identity In the identity matrix I we have aij = 0 for ij and aii = 1. Therefore the term ai11ai22...ainn is nonzero only if the permutation (i1, i2, ..., in) = (1, 2, ..., n). Thus

D(I) = sign(1, 2, ..., n) 1×1×...×1 = 1.