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



If we take a block consisting of w rows and columns, and "con- 

 dense " it, we reduce it at last to 2 terms, the first of which is the deter- 

 minant of the first 7i columns, the other of the last n columns. 



Hence, if we take the n simultaneous equations. 



and if we condense the whole block of coefficients and constants, viz. 



«M ^l,n+i 



we reduce it at last to 2 terms : let us denote them by S, T, so that 





a,, I ai,n 









, and T= 









. . .a„ 



T . 



Now we know that g, which may be written in the form 



Hence the 2 terms obtained by the process of condensation may be 

 converted into an equation for a?^ by multiplying the first of them by a?^ 

 affected with -f or — , according as n is even or odd. The latter part of 

 the rule may be simply expressed thus; — *' place the signs + and — 

 alternately over the several columns, beginning with the last, and the sign 

 which occurs over the column containing o?^ is the sign with which is to 

 be affected." 



When the vabie of cc^ has been thus found, it may he substituted in the 

 first n—i equations, and the same operation repeated on the new block, 

 which will now consist of w— 1 rows and n columns. But in calculating 

 the second series of blocks, it will be found that most of the work has been 

 already done ; in fact, of the 2 determinants required in the new block, one 

 has been already computed correctly, and the other so nearly so that it 

 only requires the last column in each of the derived blocks to be corrected. 



In the example given opposite, after writing -f- and — alternately over 

 the columns, beginning with the last, we first condense the whole block, and 

 thus obtain the 2 terms 36 and — 72. Observing that the ^-column has 

 the sign — placed over it, we multiply the 36 by — a?, and so form the 

 equation —36x= — 72, which gives x=2. 



Hence the a?-terms in the first four equations become respectively 

 2, 2, 4, and 2 ; adding these values to the constant terms in the same equa- 

 tions, we obtain a block of which we need only write down the last two 



2 4 



columns, viz. — 1 — 2 



= — 1 i6 



2 6 



