226 Dr. J. A. Fleming on the Distribution of 



Thus, for example, the solution of the three linear equations 

 ax + by +cz = d, 

 a x x + \y + 0,2 = ^1, 

 a 2 x-\rb 2 y+c 2 z = d 2 , 



is 



d b 

 do b 9 . 



c 



<>2 



X 



a x b x ci 

 a 2 b 2 c 2 



with similar expressions for y and z, differing only in having 

 as numerators respectively 



a 2 



d 

 d± 

 do 



G 



C2 



and 



a 



a x 



a 2 



b 

 h 



d 

 di 

 do 



denominator being the same. 



In this case the evaluation of these determinants is easy 

 simple symmetrical process of taking products, according 

 to the rule, 



a b c 



■ (aei + bfg + cdh) — (ecg + bdi + afh) 



icr 



d 

 9 



f 



§ 4. The properties of determinants enable us, however, 

 very easily to evaluate a numerical determinant of any order. 

 The process consists in the gradual reduction of the determi- 

 nant in order by such transformations as will render all the 

 elements of the first row or column zero except the first. The 

 determinant is then reduced to the product of its leading 

 elements and the corresponding minor. A repetition of this 

 lowers the determinant one degree at each stage ; and finally, 

 when it is resolved into a numerical two-row determinant, a 

 simple cross multiplication gives its value. 



The process of evaluation of a numerical determinant is 

 dependent on four principles : — 



(1) That the value of a determinant is not altered if rows 

 are changed into columns. 



(2) The interchange of two rows or two columns reverses 

 the sign of the determinant. 



