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)