56 BIRKHOFF AND LANGER. 



and whose determinant | Y(x) | ^ 0, is called a matrix solution of 

 equation (1). We assume at least one such solution to exist. 



Theorem: If Y(x) is any (matrix) solution of equation (1), and C 

 is any matrix of constants (for which | C | 4= 0), then Y(x), defined by 

 Y{x) = Y(x)C, is also a (matrix) solution of equation (1). 



Proof: We have ~ =— YC 

 dx dx 



dY ,(1C 



dx dx 



But by hypothesis — = A Y. 



Hence 



i.e. 



Moreover, since the product of two matrices is derived in the same 

 manner as that of two determinants, it follows that | Y | = | Y | | C \. 

 Hence | Y | =j= 0, if | C | =J= and | Y | =J: 0. Q.E.D. 



Theorem : Given any matrix of constants Y for which | }' | + 0> 

 then there exists a matrix solution of equation (1) which for any 

 preassigned x, say x = x , reduces to the matrix F<>. 



Proof: It has already been shown that when Y(x) is a matrix solu- 

 tion of equation (1), then Y(x)C is also such a solution, where C may 

 be any matrix of constants Avhose determinant is not zero. Then in 

 particular C may be chosen as the matrix } T-1 (a- ) Y , whereupon it 

 follows that Y(x) Y~ 1 (x ) Yo is also a matrix solution of equation (1). 

 This solution, however, obviously reduces to the matrix Yo when 

 x = .r . Q. E. D. 



Theorem: If Y(x) is a matrix solution of equation (1) then Y(x)C 

 is the most general solution of equation (1). 



