INDICATOR SUBSTANCES AND FLOW ANALYSIS 



625 



whereas the latter cannot be so reduced, but repre- 

 sents a set of values. In common with the differenti- 

 ating function and other functions, matrices are 

 operators and as such are subject to the algebra 

 and calculus of operators. A determinant must have 

 the same number of rows as it does of columns, but 

 matrices are not so restricted. Both determinants 

 and matrices are useful in solving simultaneous 

 equations. 



Let us consider the solution by determinants of 

 two simultaneous equations : 



jaix -\- biy = ci 

 \a2X -\- b->y = C2 



By the perhaps more familiar method of multi- 

 plying the first equation by 62 and the second Ijy hi, 

 subtracting the resulting second equation from the 

 resulting first and dividing by the new coefficient 

 of X, the solution x = (b-yCi — bic-i)/ {aib« — anh{} is 

 obtained. The same result is obtained by evaluating 

 the determinants in 



in which the denominator consists of a determinant 

 composed of the coefficients of x and v, and the 

 numerator is similar except that the coefficients 

 of X have been replaced by the column of numbers 

 constituting the right-hand members of the original 

 system (the c's). It is apparent that evaluating the 

 determinants by subtracting the product of the upper 

 right and lower left numbers from the product of 

 the upper left and lower right numbers in each case 

 leads to the same answer as was obtained above. 

 Similarly, the solution for v is obtained by replacing 

 the h column in the numerator by the c column: 



y = 



Higher order determinants, of course, require 

 somewhat more complicated patterns for their 

 evaluation, but the basic pattern is already established 

 in the 2X2 determinant. A procedure for evaluating 

 the 3X3 determinant : 



is to multiply ai by the 2X2 determinant consisting 

 of those elements not in the same row or column as 



^2 f2 



fli, I.e., 

 minus bi times 



and evaluated as described above, 

 a.. 60 



02 Ci 



plus fi times 



a-i bi 



aiib^ci — A3C.2) — biia-iC-i — asc^) + £-1(02*3 



giving 



ash-.) 



That portion of a determinant (or matrix) which 

 remains when the elements in the same row and in 

 the same column as a selected element are crossed 

 out is called the minor of that element. Evaluation 

 of 4 X 4 and higher order determinants may be 

 achieved by alternately adding and subtracting 

 the products of the elements of one row (or column) 

 by their respective minors, in extension of the pat- 

 terns described for the 2X2 and 3X3 determinants, 

 the minor of an element of a 4 X 4 determinant 

 being a 3 X 3 determinant, etc. 



When the elements selected constitute an odd- 

 numbered row (or column) (numbering from left 

 to right and from top to bottom), the first product is 

 added; if the row (or column) used is even numbered, 

 the first product is subtracted. 



The sign to be assigned to each product is given 

 by the location of the element in the following array 

 or an extension thereof: 



-I- - -I- - 



- + - + 

 + - + - 



- + - + 



Although straightforward enough in principle, 

 it is obvious that in practice evaluation of a 4 X 4 

 or higher order determinant by this procedure would 

 involve many steps and be quite a laborious task. 

 When real numbers (rather than symbols) constitute 

 the array, some simplifying procedures are avail- 

 able. Although some of these may sound unlikely at 

 first, their validity can be demonstrated, if not con- 

 clusively proved, by simple test examples. 



Some of the rules applicable to reduction of de- 

 terminants are : 



a) Multiplication of each element of one row (or 

 column) by the same number multiplies the de- 

 terminant by that number. 



h) Adding or sul^tracting the elements of one row 

 (or column) or multiples of these elements by a 



