8.2. THE DEFINITION OF THE DETERMINANT 173

Now consider the last claim. In computing sgnn (i1, · · · , iθ−1,n, iθ+1, · · · , in) there willbe the product of n−θ negative terms

(iθ+1−n) · · ·(in−n)

and the other terms in the product for computing sgnn (i1, · · · , iθ−1,n, iθ+1, · · · , in) are thosewhich are required to compute sgnn−1 (i1, · · · , iθ−1, iθ+1, · · · , in) multiplied by terms of theform (n− i j) which are nonnegative. It follows that

sgnn (i1, · · · , iθ−1,n, iθ+1, · · · , in) = (−1)n−θ sgnn−1 (i1, · · · , iθ−1, iθ+1, · · · , in)

It is obvious that if there are repeats in the list the function gives 0. ■

Lemma 8.1.2 Every ordered list of distinct numbers from {1,2, · · · ,n} can be obtainedfrom every other ordered list of distinct numbers by a finite number of switches. Also, sgnnis unique.

Proof: This is obvious if n = 1 or 2. Suppose then that it is true for sets of n− 1elements. Take two ordered lists of numbers, P1,P2. Make one switch in both if necessaryto place n at the end. Call the result Pn

1 and Pn2 . Then using induction, there are finitely

many switches in Pn1 so that it will coincide with Pn

2 . Now switch the n in what results towhere it was in P2.

To see sgnn is unique, if there exist two functions, f and g both satisfying 8.1 and8.2, you could start with f (1, · · · ,n) = g(1, · · · ,n) = 1 and applying the same sequenceof switches, eventually arrive at f (i1, · · · , in) = g(i1, · · · , in) . If any numbers are repeated,then 8.2 gives both functions are equal to zero for that ordered list. ■

Definition 8.1.3 An ordered list of distinct numbers from {1,2, · · · ,n} , say (i1, · · · , in) , iscalled a permutation. The symbol for all such permutations is Sn. The number definedabove sgnn (i1, · · · , in) is called the sign of the permutation.

A permutation can also be considered as a function from the set

{1,2, · · · ,n} to {1,2, · · · ,n}

as follows. Let f (k) = ik. Permutations are of fundamental importance in certain areasof math. For example, it was by considering permutations that Galois was able to give acriterion for solution of polynomial equations by radicals, but this is a different directionthan what is being attempted here.

In what follows sgn will often be used rather than sgnn because the context supplies theappropriate n.

8.2 The Definition of the DeterminantDefinition 8.2.1 Let f be a real valued function which has the set of ordered lists of num-bers from {1, · · · ,n} as its domain. Define ∑(k1,··· ,kn) f (k1 · · ·kn) to be the sum of all thef (k1 · · ·kn) for all possible choices of ordered lists (k1, · · · ,kn) of numbers of {1, · · · ,n} .For example,

∑(k1,k2)

f (k1,k2) = f (1,2)+ f (2,1)+ f (1,1)+ f (2,2) .