150 Rev. C. L. Dodgson on Condensation of Determinants. [May 17, 



IV. " Condensation of Determinants, being a new and brief Method 

 for computing their arithmetical values.^' By the Rev. C. L. 

 DoDGSON, M.A., Student of Christ Church, Oxford. Com- 

 municated by the Rev. Bartholomew Pkice, M.A., F.R.S. 

 Received May 15, 1866. 



If it be proposed to solve a set of n simultaneous linear equations, not 

 being all homogeneous, involving n unknowns, or to test their compatibiHty 

 when all are homogeneous, by the method of determinants, in these, as 

 well as in other cases of common occurrence, it is necessary to compute 

 the arithmetical values of one or more determinants — ^^such, for example, as 



1, 3,-2 

 2,1, 4 . 

 3,5,-1 



Now the only method, so far as I am aware, that has been hitherto 

 employed for such a purpose, is that of multiplying each term of the first 

 row or column by the determinant of its complemental minor, and affecting 

 the products with the signs + and — alternately, the determinants re- 

 quired in the process being, in their turn, broken up in the same manner 

 until determinants are finally arrived at sufficiently small for mental com- 

 putation. 



This process, in the above instance, would run thus : — - 



\tl 



= — 21-14 + 42 = 7. 



But such a process, when the block consists of 16, 25, or more terms, is 

 so tedious that the old method of elimination is much to be preferred for 

 solving simultaneous equations ; so that the new method, excepting for 

 equations containing 2 or 3 unknowns, is practically useless. 



The new method of computation, which I now proceed to explain, and 

 for which Condensation " appears to be an appropriate name, will be 

 found, I believe, to be far shorter and simpler than any hitherto employed. 



In the following remarks I shall use the word " Block *' to denote any 

 number of terms arranged in rows and columns, and " interior of a block " 

 to denote the block which remains when the first and last rows and columns 

 are erased. 



The process of " Condensation " is exhibited in the following rules, in 

 which the given block is supposed to consist of n rows and n columns : — 



(1) Arrange the given block, if necessary, so that no ciphers occur in its 

 interior. This may be done either by transposing rows or columns, or by 

 adding to certain rows the several terms of other rows multiplied by 

 certain multipliers. 



(2) Compute the determinant of every minor consisting of four adjacent 



2, I, 4 



IX 



1, 4 

 5,-1 



-2X 



3, -2 

 5.-1 



