LINEAR ALGEBRA (Math 211)
Sheet #3
DETERMINANTS

Axiomatic definition.

The determinant is a function satisfying the following axioms:

(i)
multilinear function of rows:

(ii)
skew-symmetric function of rows:

(iii)
normalization:

Examples.

Permutations. A permutation is a one-to-one correspondence between numeres 1, 2, ..., n.

Example. All permutations of {1, 2, 3} are (1, 2, 3) (1,3,2) (2,1,3) (2,3,1) (3,1,2) (3,2,1).

Product:

Sn is the group of all permutations of the set .

An inversion in a permutation is a pair i,j such that i<j but .

A sign of a permutation.