Br Baker, On the Invariant Factors of a Determinant 73 



We desire to shew now that, in the notation of § 5, the first k 

 of the series of exponents 



e i > € 2 > € 3 , ... 



are each equal to a, the succeeding jjl of them each equal to /3, 

 and so on, and finally the last a- of them each equal to <y. It will 

 then follow from the preceding work that 



e l > e 2 > €3 > . . . . 



As we have remarked, the invariant factors are the same for the 

 matrix A — p as for the matrix a — p ; it is then sufficient to prove 

 the result for A — p. 



\ 7. For this purpose let the elements of A — p be denoted 

 by b{j and the matrix itself by b ; let cp be an arbitrary matrix of 

 n rows and columns of non- vanishing determinant. 



The expansion of the determinant of the matrix b + - (f> in 



v 



powers of - being 



\b\ + - t 8\b\ + l^\b\ + ..., 



where 8 = 22 cpij ~r- , 



we see that the first term will divide by (p — 6) l \ the second term, 

 being a linear aggregate of first minors of |6|, each multiplied by 

 an element of <£, will divide by (p — d) li \ similarly the third will 

 divide by (p — 6) 1 *, and so on ; if, as before, l r -i > 0, l r = l r+1 = . . . = 0, 



the term in (-] , being 



and therefore an aggregate of determinants of minors of |6| of 



n — r rows and columns, will contain one term at least not divisible 



by p — 6, and therefore will not divide by p — 0; while the term in 



ly-i ... z 



will divide by (p — 6) r ~\ 



t. 



Thus if p — 6 be denoted by ®, we have an equation of the 

 form 



\(f> + tb\ = t n ® l H + ^- 1 © Z - £ 'ZT 1 + . . . + t n - r+1 0^-...-^,^ + ^ 



