54 PROCEEDINGS OF THE AMERICAN ACADEMY. 



contained in B' but not in B with their correspondents of the same 

 suffix, this term will pass over into a term of the complement of M. 

 In this process C and D are not changed for each an. in C is either not 

 changed at all or replaced by the same letter and no letter of D can be 

 in a column so moved. Furthermore the sign is correct for there are 

 as many minus columns introduced as interchanges made. The same 

 argument shows that for every product of minors in the algebraic 

 complement of M there is an equal product in M. Therefore M is 

 equal to its algebraic complement. 

 Suppose now p -\- q > n. Then 



71 — p-\- n — q -^ n. 



We expand the determinant A in terms of minors of the nth. order 

 taken from the first n and the last n columns. The part of the ex- 

 pansion Avhich contains all the a's and jS's in the minor from the first 

 n rows is 



Aj = — {Ai A2. . . -An^q Bi B2. . . -Bg) {An-q+l. . ■ -Ap ttp^i a„ 



/?g+l . . . -l^n) 



the summation being for every combination of n — q A's in the first 

 factor with the remaining p -\- q — n in the second factor so arranged 

 that the two groups in the order written constitute a positive permuta- 

 tion of Ai to Ap. The form of this expression is evident since the 

 5's cannot occur in the same factor with a's and jS's (the other factor 

 then containing a row of zeros). The sign of the term written is 

 positive since it is obtained as a product of principal minors given 

 by moving q rows of 5's past 11 — p rows of a's and p + q — n rows 

 of ^'s, interchanging first n and last n columns and changing n — q 

 minus signs. The result should therefore have a sign 



( l'\q{n—p+p+q—n)+)r-+n — q =2. 



The signs of the other terms then follow, since any positive rearrange- 

 ment of A's should not change the sign of the term. 



Now in the expression of A each minor formed of n — p rows of a's 

 and n — q rows of jS's is equal to its coefficient. Furthermore Aj 

 contains all of the terms in A given by such minors taken from the 

 first 71 columns. Therefore in A^ each minor of the matrix 

 [ap+i...a„ |8g+i.../3„] is equal to its coefficient. These coefficients 

 constitute the matrix 



^ {A]. . .An-q Bx . . .Bg) [An^+i . . . .A^ 



