Consider a general n by n matrix
A = [ | a_{11} | a_{12} | ... | a_{1n} | ] = [a_{11}e_{1} + a_{21}e_{2} + ... + a_{n1}e_{n}, a_{12}e_{1} + a_{22}e_{2} + ... + a_{n2}e_{n}, ...]. |
a_{21} | a_{22} | ... | a_{2n} | ||
: | : | : | |||
a_{n1} | a_{n2} | ... | a_{nn} |
Our detailed computation for multilinear and alternating functions of 2 by 2 and 3 by 3 matrices give all the clues on the general formula for detA. The multilinearity implies
detA = ∑_{1≤i1,i2,...,in≤n} a_{i11}a_{i22}...a_{inn} det(e_{i1}, e_{i2}, ..., e_{in}).
By the alternating property, if any two of the indices i_{*} are the same, then det(e_{i1}, e_{i2}, ..., e_{in}) = 0. Thus only the terms with distinct i_{1}, i_{2}, ..., i_{n} remain
detA = ∑_{i1,i2,...,in distinct} a_{i11}a_{i22}...a_{inn} det(e_{i1}, e_{i2}, ..., e_{in}).
The following facts
imply that (i_{1}, i_{2}, ..., i_{n}) fits into the following definition.
A permutation (i_{1}, i_{2}, ..., i_{n}) of (1, 2, ..., n) is a rearrangement of positions of the numbers. Equivalently, the map k → i_{k} is an invertible map from {1, 2, ..., n} to itself. The sign of the permutation is
For example, the permutation (3, 1, 4, 2) represents the following invertible map from {1, 2, 3, 4} to itself:
1 → 3, 2 → 1, 3 → 4, 4 → 2.
Moreover, the following sequence of transpositions
(3, 1, 4, 2) → (1, 3, 4, 2) → (1, 3, 2, 4) → (1, 2, 3, 4)
tells us sign(3, 1, 4, 2) = -1. Correspondingly,
det(e_{3}, e_{1}, e_{4}, e_{2}) = - det(e_{1}, e_{3}, e_{4}, e_{2}) = det(e_{1}, e_{3}, e_{2}, e_{4}) = - det(e_{1}, e_{2}, e_{3}, e_{4}) = -1 = sign(3, 1, 4, 2).
where the red equality follows from the third condition in the definition of the determinant.
The discussion works in general. In particular, even (odd) number of transpositions will change sign even (odd) number of times and give us positive (negative) sign at the end. Thus we have
det(e_{i1}, e_{i2}, ..., e_{in}) = sign(i_{1}, i_{2}, ..., i_{n}).
Now we are able to write down the explicit formula for the determinant.
detA = ∑_{all permutations (i1,i2,...,in) of (1,2,...,n)} sign(i_{1}, i_{2}, ..., i_{n}) a_{i11}a_{i22}...a_{inn}.
We remark that, if we did not use the condition detI_{n} = 1 in the red equality above, then we get the following conclusion.
D is a multilinear and alternating function of n by n matricies ⇒ D = d det for a constant d = D(I).