We wish to prove that the plus sign should be used in (9) if both of the permutations of (10) are even or both are odd, and that the minus sign should be used when one is even and the other odd. When the entries in (9) have their row numbers in the normal order, the permutations (10) are of the form

with the top one even, and hence the new 2-permutation rule indicates that the sign should be plus when the bottom permutation is even and minus otherwise. This agrees with the definition of a determinant and shows that the new rule is correct in this case.

We can go from the order a_1ia_2j...a_nh of the factors to any order a_pua_qv...a_rw by means of a number of interchanges of adjacent factors. Whenever one such interchange is made, the row subscript permutation and column subscript permutation will each change from even to odd or from odd to even. This means that the new rule will continue to indicate the correct sign as these interchanges are made. We are especially interested in the case in which the column numbers are in order, that is, the case in which (9) is (8)

The permutations (10) then become
 

 
with the bottom one even. Hence the 2-permutation rule indicates that the plus sign is used if and only if the permutation x, y, ..., z of (11) is even. This is exactly the rule for determining the sign of (8)
 

 
as a term of D'. Hence the sign for (8) is the same either as a term of D' or of D, since the sign in both cases agrees with the 2-permutation rule. This shows that D' = D and completes the proof.
 
 

Tuesday, June 23, 1998