Chapter 20

Theory of Determinants∗

You might skip this chapter and return to it later if accepting the outrageous claimsabout the determinant, that it is independent of the row or column chosen, does not botheryou. If this does cause some cognitive dissonance, then you should read this chapter now.

20.1 The Function sgnThe following Lemma will be essential in the definition of the determinant.

Lemma 20.1.1 There exists a function, sgnn which maps each ordered list of numbersfrom {1, · · · ,n} to one of the three numbers, 0,1, or −1 which also has the following prop-erties.

sgnn (1, · · · ,n) = 1 (20.1)

sgnn (i1, · · · , p, · · · ,q, · · · , in) =−sgnn (i1, · · · ,q, · · · , p, · · · , in) (20.2)

In words, the second property states that if two of the numbers are switched, the value ofthe function is multiplied by −1. Also, in the case where n > 1 and {i1, · · · , in}= {1, · · · ,n}so that every number from {1, · · · ,n} appears in the ordered list, (i1, · · · , in) ,

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

(−1)n−θ sgnn−1 (i1, · · · , iθ−1, iθ+1, · · · , in) (20.3)

where n = iθ in the ordered list, (i1, · · · , in) .

Proof: Define sign(x) = 1 if x > 0,−1 if x < 0 and 0 if x = 0. If n = 1, there is onlyone list and it is just the number 1. Thus one can define sgn1 (1)≡ 1. For the general casewhere n > 1, simply define

sgnn (i1, · · · , in)≡ sign

(∏r<s

(is − ir)

)This delivers either −1,1, or 0 by definition. What about the other claims? Suppose youswitch ip with iq where p < q so two numbers in the ordered list (i1, · · · , in) are switched.Denote the new ordered list of numbers as ( j1, · · · , jn) . Thus jp = iq and jq = ip and ifr /∈ {p,q} , jr = ir. See the following illustration

i11

i22

· · · ip

p· · · iq

q· · · in

n

i11

i22

· · · iqp

· · · ip

q· · · in

n

j11

j22

· · · jp

p· · · jq

q· · · jn

n

431