 
Summary: Lecture 21: Remarks on permutations and determinants
1. Permutations, Existence and uniqueness of the determinant
A permutation of the set {1, . . ., n} is a rearrangement of the sequence (1, . . . , n).
The following list contains all permutations of the set {1, 2, 3}. There are 3! = 6 of
them
(1, 2, 3) , (1, 3, 2) , (2, 1, 3) , (2, 3, 1) , (3, 1, 2) , (3, 2, 1).
The formally correct definition is the following: A permutation of the set {1, . . ., n}
is a function
: {1, . . . , n}  {1, . . ., n}
such that j = k implies (j) = (k). For example, the permutation (1, 3, 2) in the
above list corresponds to the function
(1) = 1 , (2) = 3 , (3) = 2.
Some permutations simply switch two elements of the sequence, such as the per
mutation (3, 2, 1). We will call them 'simple permutations'. If ((1), . . . , (n)) is a
permutation, and if N simple permutations applied successively transform it back
to (1, . . . , n) then (1)N
is called the sign of the permutation . Here are the signs
of the permutations of the set {1, 2, 3}
permutation sign
(1, 2, 3) +1
