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ME. H. J. S. SMITH ON SYSTEMS OF LINEAE 
Art. 8. From the preceding principles we may deduce the solution of the follomng 
problem, which admits of important applications in other parts of arithmetic : — 
“To find all the matrices of a given type, of which the determinants have given 
^alues, not all equal to zero.” 
Two particular cases of this problem (those in which the matrix is of the type 2x3 
and 2x4) occur in the ‘ Disquisitiones Arithmeticse ’ (see arts. 279 and 236). In both 
places Gauss has suppressed the analysis of the problem, and has only given a synthe- 
tical demonstration that its conditions are satisfied by the solution he assigns. This, 
indeed, in art. 279, he expressly observes. He has also suppressed his method of 
deducing the complete solution from any particular solution, — an omission, however, 
which may probably be supplied by a comparison of art. 234 with art. 213, i. The veiy 
general and most important case, of a matrix of the type nX{n-\-l), has been subse- 
quently treated of by M. Hermite 
Let ||a|| denote a matrix of the type nx{n-^'ni), of which the constituents are abso- 
... .14 
lutely indeterminate quantities ; writing X for we shall represent its determi- 
nants by Xi, X 2 , . . . . X;^. . If w>l, these determinants are not all independent, but 
are connected by certain identities of the form 
<h(X,X„....XJ=0, (29.) 
denoting a rational and integral homogeneous function with numerical coefficients. 
If, therefore, Ci, C 2 , ... be a given set of integral numbers, which can be represented 
as the determinants of a matrix of the type nx{n-\-m), these numbers will satisfy every 
relation of the form (29.); so that the identity 
0(X„ X 2 , ...XJ=0 
Avill involve also the numerical equation 
0(C., C 2 , ...CJ=0 (30.) 
To obtain a convenient notation for Cj, C 2 , . . . let us imagine that we have formed 
a square matrix of the type { 71 -{-m) X {7i-\-m) by the addition of m horizontal rows to the 
matrix |ja:|| ; if, in the development of the determinant of this matrix, the coefficient of 
Xj be the determinant 
r=l, 2, 3, ... m 
^n + r, fXg ^ ^ 
s = l, 2, o, . . . m 
(///j, denoting 7n of the numbers 1, 2, . . . n-\-m), we may represent X,- and C, 
by the symbols [jzj, . . . yj^] and [jj^, . . . respectively ; observing, however, that 
if two of the numbers jt-i, yj^-, • ■ • are equal, the value zero is to be attributed to each of 
these symbols. 
If r denote one of the numbers 1, 2, 3, . . . %, the determinants of the matrix obtained 
* Ceelle’s Journal, vol. xl. p. 264 ; see also Eisensteik, ibid. vol. xxviii. p. 327 . 
