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:

**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:**

*S*_{n} 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.**