37. Laplace's expansion rule

We will now generalize the arguments of Section 10, which led us to the expansion formulae. Let r be an integer in the interval 0 < r < n amd s = n - r. In the determinant

collect all those terms in which the numbers p₁, ··· , p_r, apart from their sequence, equal 1, ··· , r.The corresponding permutations P* of 1, ··· , n are such which decompose into a permutation P₁ = [p₁, ··· , p_r] of the numbers 1,···,r and a permutation P₂ = [p_r+1, ··· , p_n] of the numbers r+1,···,n. It is readily seen that c(P*) = c(P₁)c(P₂), because all inversions of P* are formed by p₁, ··· , p_r on their own and by p_r+1, ··· , p_n on their own, while inversions between p₁, ··· , p_r and p_r+1, ··· , p_n are not possible, because all the numbers of the first partial set are smaller than those of the second partial set. As P* passes through all permutations of the type considered, P₁ and P₂ pass through all permutations of 1,···,r and r + 1, ··· , n.

The intended arrangement yields

Now let, in general, quite a = {a₁, ··· , a_r} and b= {b₁, ··· , b_r} be two partial sets of the numbers 1, ··· , n, each with r numbers and arranged according to their size, that is, a₁ < a₂ < ··· < a_r and b₁ < b₂ < ··· < b_r.

Denote by D_ab those minor determinants of order r of |A|, which arise by omission of all rows and columns, the numbers of which do not occur in the sets a and b, respectively.

Denote by the complementary sets of a and b with respect to 1, ··· , n so that and each together account for the sequence of the numbers 1, ··· , n. Again let and . The minor determinant of order s will be called the minor determinant, complementary to D_ab. For example, the two minor determinants in (37.2) complement each other.

Determine now the factor by which the minor determinant D_ab. in |A| is multiplied. For this purpose, we interchange in |A| rows and columns in such a manner that the rows with the numbers a₁, ··· , a_r move to the top and the columns with the numbers b₁, ···, b_r. to the left. In order to move Row a₁ to the first position, you require a₁- 1 interchanges with rows above; in order to shift then Row a₂ to the second position, you require a₂ - 2 interchanges, etc. Altogether you require a_r- r interchanges, in order to shift Row a_r to the r-th position. Hence the proposed reordering of the row demands altogether

a = a₁- 1 + a₂ - 2 + ··· + a_r- r = a₁ + a₂ ··· + a_r- r(r + 1)/2

interchanges of two neighbouring rows. In the process, |A| is multiplied by (-1)^a. If you next rearrange the columns in a corresponding manner, |A| is multiplied in addition by (-1)^b, where

b = b₁ + ··· + b_r- r(r + 1)/2.

In the reordered determinant with the value

is D_ab at the top on the left hand side. Then, according to what has been stated above, D_ab in (-1)^a+b|A| is multiplied by the complementary minor determinant . Hence, D_ab appears in A itself with the factor

.

A_ab is called the adjoint minor determinant of D_ab.

Out of the rows a₁, ··· , a_r, you can form altogether minor determinants of order r. Denote these in an arbitray sequence by D₁, ··· , D_m. As we have just seen, there appears in |A| every D_i with its adjoint minor determinant A_i, whence

This is Laplace's expansion of |A| with respect to the rows a₁, ··· , a_r. There exists also a corresponding expansion with respect to r columns. In the case r = 1, you arrive at the expansion formulae of Section 10.

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