Dr Baker, On the Invariant Factors of a Determinant. 71 



or the set also satisfies (a — 6) r x = 0. From this it follows that if 

 z be a set of solutions of {a — 6) l+1 z = or c (a — 6) z = 0, then 

 (a-6) r+1 z = 0. 



Suppose now if possible that \ < I ; then p — 6, which occurs 

 in \a — p\ to power I, must divide \k — p\; and hence a set, u, of 

 n — A, quantities can be found so that ku = 6u ; then, by car) = cr}/c, 

 we have cawu = cq6u, or (a — 6) l+1 rju = 0, and hence (a — d) r+1 nu = 0, 

 and therefore, as r^\<l, also (a — 6) 1 rju = 0, or crju = 0, shewing 

 that rju is capable of a form gt, where t is a set of A, quantities ; 

 but the equation (£, v) (t, — u) = or <r(t, — u) = is contrary to 

 the hypothesis that a is of non-vanishing determinant. 



§ 5. Hence by § 3 we can form a matrix M of non-vanishing 

 determinant of n rows and columns of which the first I columns 

 are formed by the constituents respectively of the sets w {1) , ..., w {l) , 

 arising for D l x — 0, the following I' columns are formed by the 

 constituents respectively of the sets w' {1) , ..., w' (V) , arising for 

 (a — d') v x = 0, and so on. Since equations 



are the same as 



aw {i) = 0w {i \ aw {i+1) = w^ + 0w {i+1) , 

 etc., what we have proved is equivalent to 



aM=MA, 



where J. is a matrix having elements only in the diagonal con- 

 taining the places (1, 1), ..., (n, n) and in the line parallel thereto 

 and next to it on the right which contains the places (1, 2), (2, 3), 



(3, 4), The elements in the diagonal of A consist of 6 coming 



I times, followed by 6' coming V times, and so on ; the elements 

 in the other line consist of sequences of unities separated by 

 zeros, the number and position of the zeros being determined by 

 the scheme in § 2. Since therein some of the numbers f, g —f, ... 

 may be zero, it is desirable to rewrite it, with the omission of 

 zeros, say as follows : 





D Vl M = 0, D Vs v = Vl w, ... , Drff = v f_ 



k rows, a columns, 



p, rows, j3 columns, 



