626 



HANDBOOK OF PHYSIOLOGY 



' CIRCULATION I 



constant to or from the corresponding elements of a 

 second row (or column) and replacing the second 

 row (or column) by the resulting sums or differences 

 does not change the value of the determinant. 



c) Exchanging the positions of any two rows (or 

 columns) changes the sign of the determinant but 

 not its absolute value. The sign resulting when the 

 sequence of rows (or columns) is altered may be 

 determined from the number of exchanges of pairs 

 of rows (or columns) required to achieve the same 

 end result. 



d) If all of the elements except one of a row (or 

 column) are zero, the other members of its column 

 (or row) may be changed to zero without changing 

 the value of the determinant. 



By application of the above rules, a given de- 

 terminant may be reduced to one which contains all 

 zeros except along a principal diagonal. The following 

 example was constructed to illustrate the use of the 

 above rules : 



1000 

 1002 

 3 4 3 4 

 3030 



(2) 



000 

 002 

 400 



030 



= (-2) 



= (2) 



I o o 

 020 



003 



000 



= (2)(24) = 48 



In the first step, each member of row 2 is divided by 

 2, so 2 becomes a coefficient of the determinant. 

 In the second step the first column is subtracted 

 from each of the other three columns. The third 

 step could be achieved by subtracting multiples of 

 the top row from the other rows, but can also be 

 achieved immediately using rule d, which may also 

 be used for the next step. Finally, the column order 

 and the sign are changed and the answer is the 

 product of the numbers on the diagonal and the 

 coefficients of the final determinant. 



Returning to the subject of matrices, a few special 

 properties may be noted. In particular, two m X n 

 matrices may be added or subtracted by performing 

 the corresponding operation on the pairs of elements 

 having corresponding locations. That is, (a,j + 6„) = 



Cij where the subscripts i and j denote the location 

 by row and column number. Multiplication of two 

 matrices requires that the first have the same number 

 of columns as the second does of rows, for in matrix 

 multiplication each element of a row of the first is 

 multiplied by the corresponding (sequentially) ele- 

 ment in a column of the second, and the sum of these 

 products is entered as the element in the product 

 matrix at the location where the row and column 

 used intersect. Thus, 



Va b~\ Vh k~[ _ Vah + bv ak + bv~\ 

 \_c d\ ' \_w i\ \_ch + dv ck + dv] 



The result is different if the first matrix is exchanged 

 for the second, and this is one of the respects in which 

 matrix multiplication differs from ordinary algebraic 

 multiplication. The two cases of matrix multiplication 

 are called pre- and post-multiplication (by the factor 

 involved). 



Division by matrices is not defined. Thus, to solve 

 the matrix equation 



[.4] [A-] = [B] 



for [A'], it is not possible to divide both sides by [/I]. 

 The desired result is achieved, however, by multi- 

 plying both sides by the inverse of [_A], \_A'\~^. To 

 understand the inverse matrix concept, it is first 

 necessary to know about the identity matrix. As in 

 algebra multiplication by unity does not change the 

 value of a nuinber, and multiplication of a number 

 by its inverse (in this case, reciprocal) gives unit^', 

 so in matrix algebra the identity matrix is defined as 

 that matrix which, when used as a multiplier, does 

 not change the value of the matrix operated on, and 

 the inverse matrix is defined as being the one which, 

 when multiplied by the original matrix, yields the 

 identity matrix. The identity matrix turns out to be 

 the unit diagonal inatrix, for example: 



I 



I o o 

 O I o 



O O I 



One of the several available methods for generating 

 the inverse of a matrix is illustrated in a later section 

 of this chapter. 



THE LAPLACE TRANSFORM. The Laplacc transform is a 

 mathematical device which is useful in simplifying 

 the method of solution of many differential equations, 

 and is particularly suitable for the solution of ordinary 

 differential equations with constant coefficients. 



