1866.] Rev. C. L. Dodgson on Condensation of Determinants . 153 



The theorem referred to is the following : — 



** If the determinant of a block = R, the determinant of any minor of 

 the mth. degree of the adjugate block is the product of R'""^ and the 

 coefficient which, in R, multiphes the determinant of the corresponding 

 minor.'* 



Let us first take a block of 9 terms. 



__ _ =R; 



1 «3,2 «3,3 



and let a^^ , represent the determinant of the complemental minor of a^^ ^, 

 and so on. 



If we " condense " this, by the method already given, we get the block 

 J ^3.3 ^j.i I ^ and, by the theorem ^bove cited, the determinant of this, 



Rx a„ „. 



R: 



Hence 



Secondly, let us take a block of 1 6 terms : 

 a, a. 



which proves the rule. 



= R. 



If we "condense" this, we get a block of 9 terms ; let us denote it by 



, in which h^^^ = 



2, 1 ^2, 2 



&c. 



If we "condense" this block again, we get a block of 4 terms, each 

 of which, by the preceding paragraph, is the determinant of 9 terms 



of the oridnal block; that is to say, we get the block | ^^-^ ^4,1 1 

 but, by the theorem already quoted, 



""'•^ ==Iix5,,; therefore 



R 



'4,4 "^4,1 



that is, R may be obtained by "condensing" the block 



^2,2 



f a4,4 



lai,4 ^1,1 J' 



This proves the rule for a block of 16 terms ; and similar proofs might 

 be given for larger blocks. 



I shall conclude by showing how this process may be applied to the 

 solution of simultaneous linear equations. 



