THOMAS MUIR ON GENERAL THEOREMS ON DETERMINANTS. ol 
If the excess of the number indicating the order of the original determinant 
over that indicating the order of a minor of it be E, it is readily seen that the 
number of possible expansions thus obtainable for the product of the determi- 
nant and its minor is the highest integer in 4 E. 
2. REDUCTION OF THE ORDER OF A DETERMINANT.—Taking the determinant 
Ue a we 
Gage Wa Mijn as Dy, 
Cesk ie en |,.00 Di Gin), 
Grr Gn2 Uz + + + Un 
and multiplying each element of the first column by —a,, and adding to the 
result @,, times the corresponding element of the second column, we have 
0 Ogi Gig». «-» Gin 
Wis Cog | doo. Eons - » Coy, 
Ol) | | Cay Gen] Oon Doge. - - Can 
aa One | One Ong aE Sin se Gan 
Similar operations lead finally to 
(1) ayy yg +. 
Hence, dividing by (—1)""a,, 
1 Ca) A43 ef he Qn 
Ao Ag9 M53 Te octets Aon 
431 Azo Ugg . . « Azn 
) 0 ie ) Gn 
Vay Goo | tala Cowl eaten | Crest Gay dl oben 
din D(Qin) = | | yy Ago | | Aq %g| - - - | @in-1 Gan | sn 
| Ay M2 | | A495 Ung | i ae a8 | Un-1 Dan, Ann 
a 
@yg... Ain, we have 
[241 Gq] [@y9 Gog] - - + [@in—1 Gen] 
ihe 1 241 Uo] 19 Ugg] ~~ - [@in—1 Gon 
CO SS GS ie) py Cd 
ay Ang| a4 Ans| Oe \@1n—1 Un 
the second determinant being of the (n—1)* order. 
This identity has been long known: it is proved in Brioscut by means of 
VOL, XXIX, PART I. 
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