SYSTEMS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS. 



167 



Hence 



A'=iliV • • • • Km'HKxH! 



K m |/ A "■ = A'" - 7 A" 1 = i/A 



Now, since £/, , k„/ are all integral functions of D , A' must be an integral 



function of D, and therefore also l/A ; but this is impossible unless A reduce to a con- 

 stant. Hence the condition is not only sufficient but necessary ; and we have now the 

 theorem in the form — 



When tivo systems of linear equations -with constant coefficients are equivalent, the 

 modulus of either with respect to the other must be constant, and the converse is also true. 



This equivalence theorem can be expressed in another form, which is convenient for 



some purposes. 



The matrix 



(11) (12) (In) 



(ml) (m2) . 



{run) 



(10), 



whose m rows are made up of the operator coefficients of the dependent variables in the 

 m equations of a system, may be called the Matrix of the System. 

 From (7) we have 



[11] = (11)&+(21)£+ +(ml)| m 



[In] = (ln)g 1 +(2n)g 2 + + (mn)g„ l 



[21] = (11)7 ?1 + <21>,,+ +(ml) Vm 



[2n] = (ln) >ll + (2ii)>i 2 + +( / nin)ij m 



&c, &e. 



Hence we have 



[11] 

 [ml] 



[In] 

 [mri] 



(11) 

 (ml) 



(In) 



(inn) 



(11) 



in the sense that every determinant in the first matrix is equal to the corresponding 

 determinant in the second, multiplied by A . Hence 



The necessary and sufficient condition that two systems of linear equations with con- 

 stant coefficients be equivalent, is that every determinant in the matrix of the one system 

 differs by the same constant multiplier from the corresponding determinant in the matrix 

 of the other. 



In particular, 



In order that two determinate systems {of n equations in n dependent variables) be 

 equivalent it is necessary and sufficient that the determinants of the two systems differ 

 merely by a constant factor — that is to say, that 



1 (11) (22) .... (nn) | = | [11] [22] .... [nn] | x const. 



