math111_logo Theory of Determinant

2. General formula

Consider a general n by n matrix

A = [ a11 a12 ... a1n ] = [a11e1 + a21e2 + ... + an1en, a12e1 + a22e2 + ... + an2en, ...].
a21 a22 ... a2n
: : :
an1 an2 ... ann

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,...,inn ai11ai22...ainn det(ei1, ei2, ..., ein).

By the alternating property, if any two of the indices i* are the same, then det(ei1, ei2, ..., ein) = 0. Thus only the terms with distinct i1, i2, ..., in remain

detA = ∑i1,i2,...,in distinct ai11ai22...ainn det(ei1, ei2, ..., ein).

The following facts

  1. the total number of indices is n,
  2. indices are integers between 1 and n,
  3. indices are distinct,

imply that (i1, i2, ..., in) fits into the following definition.

A permutation (i1, i2, ..., in) of (1, 2, ..., n) is a rearrangement of positions of the numbers. Equivalently, the map kik 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(e3, e1, e4, e2) = - det(e1, e3, e4, e2) = det(e1, e3, e2, e4) = - det(e1, e2, e3, e4) = -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(ei1, ei2, ..., ein) = sign(i1, i2, ..., in).

Now we are able to write down the explicit formula for the determinant.

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

We remark that, if we did not use the condition detIn = 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).


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[Extra: The sign of permutations is well-defined]