5.3. THE MATHEMATICAL THEORY OF DETERMINANTS 67

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) (5.3.6)

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

nThen

sgnn ( j1, · · · , jn)≡ sign

(∏r<s

( js− jr)

)

= sign

 both p,q(ip− iq)

one of p,q︷ ︸︸ ︷∏

p< j<q(i j− iq) ∏

p< j<q(ip− i j)

neither p nor q

∏r<s,r,s/∈{p,q}

(is− ir)

The last product consists of the product of terms which were in the un-switched product∏r<s (is− ir) so produces no change in sign, while the two products in the middle bothintroduce q− p−1 minus signs. Thus their product produces no change in sign. The firstfactor is of opposite sign to the iq− ip which occured in sgnn (i1, · · · , in) . Therefore, thisswitch introduced a minus sign and

sgnn ( j1, · · · , jn) =−sgnn (i1, · · · , in)