Dr Baker, On the Invariant Factors of a Determinant. 69 



D v (p+d = q } Du { p+v = v^ +1) ) 



\ q — p rows, 2 columns, 



Dv® = 0, Du® = v® J 



2) W (?+D =0] 



L r — g rows, 1 column, 



D»« = 0j 



wherein any of the quantities f, g — f, ..., q — p, r — q may- 

 be zero. 



§ 3. Denote /+ ... + r by X and these sets of solutions corre- 

 sponding to the factor (p — 6) 1 of 



F(p) = \a-p\ = (-ir( P -ey(p-ey... 



by w {1) , ..., w {K) . Corresponding to the other factors {p — 9') l \... 

 we shall similarly obtain lots of sets of solutions 



w'u, ..., w' {K \ w"^, .... 



These X + X' + . . . sets are linearly independent ; for the result of 

 operating on any supposed linear equation 



P 2 wW + . . . + P x t0W + Pi W'^ + ...+ P\-W'^ + P/V'W + . . . = 



with the matrix 



®(a) = (a-dy'(a-9"y" ... 



is to give 



<f> (a) [P>w + . . . + P K v)W] = ; 



we have however 



(a - Of [P lW v + ...+ P K w^] = 



and polynomials in the matrix a, <J>j (a), "^ (a), can be found to 

 make 



$! (a) $ (a) + ¥ (a) (a - 0)* = 1, 



and therefore also 



PjlwW + ... + P a w< a »=0, 



contrary to the proved linear independence of w (1) , ..., w (A) . 



If then we can prove that the numbers X, \', ... are respectively 

 at least as great as I, l', ..., we shall have at least I + I' + ..., = n, 

 linearly independent sets of solutions arising altogether ; since of 

 sets of n elements at most n are linearly independent we shall 

 therefore have just n. Thus the proof that the numbers X, X', ... 

 are respectively at least as great as I, V, ... involves that X = I, 



x' = r, .... 



