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



a rows, 7 columns, 



where l^a>{3>. ..>y>0, k>0, //, > 0, . . . , <r > 0, 



/ca + /a/3 + . . . + gt7 = £. 



With this notation, the matrix A in the equation 



M~ 1 aM = A, 



will have the first a — 1 elements (1, 2), (2, 3) ... each unity, then 

 the element (a, a + 1) a zero, then the next a — 1 elements 

 (a + 1, a + 2), ... unities followed by a zero, and so on. 



Further this equation gives, if p be an arbitrary single 

 quantity, 



M~ x (a - p) M = A - p 



and therefore \a — p\ = \A — p\. It is however the case that the 

 power of a factor p — 8 which is common to all minors of the 

 determinant \a— p\ of s rows and columns is the same as the 

 power common to all minors of \A — p\ of the same order; for 

 suppose these powers to be respectively p s and P s ; then from 



A - p = if" 1 (a - p) M, a- p = M{A- p) M~\ 



since the determinant of a minor, of the sth order, of a product 

 of two matrices is a linear aggregate of products of determinants 

 of minors, also of the sth order, from the component matrices, it 

 follows, respectively, that 



■* s > Ps> Ps > "s- 



§ 6. Now supposing as before (p — 6) 1 to be the highest power 

 of a factor p — 6 of the determinant | a — p | , let (p — #)*• be the 

 highest power of the same factor dividing all minors of this 

 determinant of (n— 1) rows and columns, (p — 6) 1 * the highest 

 power dividing all minors of (n — 2) rows and columns, and so on, 

 and put 



6x = v — li, € 2 = L\ — t 2 » • • • > e r == "r— 1> e r+i = = e r -j- 2 = • • • 



so that n — r is the rank of the matrix a — p corresponding to 

 p — 6; then 



(p-ey = (p-ey>(p-dy>...(p-dyr 



and (p — #)% (p — dy* . . . are called the invariant factors of the 

 matrix a — p or of the determinant |a — /o| corresponding to the 

 root 6. 



