Application

Permutations and the sign of permutations occur in the explicit expression for determinants. If A is an n by n matrix with entries A_i,j then the determinant det(A) is the sum over all n! permutations g in S_n of the products

sgn(g) A_1,g(1) A_2,g(2) ··· A_n,g(n),

i.e.,

det(A) = _g sgn(g) A_1,g(1) A_2,g(2) ··· A_n,g(n).

In the case of a 2 by 2 matrix A = [[A_1,1, A_1,2],[A_2,1, A_2,2]] we find two terms:

• A_1,1A_2,2 corresponding to the identity permutation, which has sign 1, and
• -A_1,2A_2,1 corresponding to the permutation (1,2), which has sign -1.

Summing yields the familiar formula

det(A) = A_1,1A_2,2 - A_1,2A_2,1.

It is still easy to write down the explicit 6 term formula for a 3 by 3 determinant, but since n! grows so rapidly, the formula becomes quite impractical for computations if n gets large. For computations of determinants more practical methods are available derived from the above formula. Such methods are discussed in courses on linear algebra.