The following theorem enables one to prove results involving columns of a determinant from the corresponding results on the rows and vice versa.
 

THE TRANSPOSE THEOREM: Let D' be the transpose of a determinant D. Then D' = D.
 

Proof: Let a_rs and b_rs be the entries of D and D', respectively, in the rth row and sth column. Since D' is the transpose of D, b_rs = a_sr. By definition, D is the sum of n! terms

where the plus sign is used if the permutation
 

is even and the minus sign if it is odd. Also, D' is the sum of n! terms

where the sign is plus if

is an even permutation, and minus otherwise. Since b_rs = a_sr, each term (6) can be rewritten as

The terms (8) are all the products, with signs attached, in which there is exactly one factor from each row and from each column of D; hence the terms (8) are the terms of the expansion of D, except that the signs may not agree. We will therefore prove that D = D' by showing that (8) has the same sign as a term of the expansion of D that it has in the expansion of D'. We do this by describing a method of determining the sign whether or not the row numbers of the entries are in the order 1, 2, ..., n. Let

be a term of the expansion of D with its factors in any order. Then the row subscript numbers and the column subscript numbers give us the two permutations:

Wednesday, June 10, 1998