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XV. On Systems of Linear Indeterminate Equations and Congruences. By Henky J. 
Stephen Smith, Fellow and Mathematical Lecturer of Balliol College, Oxford. 
Communicated by J. J. Sylvester, Esq., F.B.S. 
Eeceived January 17, — Eead January 31, 1861. 
The theory of the solution, in positive or negative integral numbers, of systems of linear 
indeterminate equations, requires the consideration of rectangular matrices, the con- 
stituents of which are integral numbers. It mil therefore be convenient to explain the 
meaning which we shall attach to certain phrases and symbols relating to such matrices. 
A matrix containing constituents in every horizontal row, and q in every vertical 
column, is a matrix of the type qXp. We shall employ the symbol 
qxjy 
A 
, or (when 
it is not necessary that the type of the matrix should be indicated in its symbol) the 
simpler symbol ||A[| to represent the matrix 
-'^1, n -^1, 7.1 • • 
n -'^2, 21 • • 
A A 
. . . A 
If |;A|| and jjB[j be two matrices of the same type, the equation ||A|| = ||B|| indicates that 
the constituents of ||A|| are respectively equal to the constituents of ||B|| ; whereas the 
equation jAj=|B| will merely express that the determinants of ||A|| are equal to the cor- 
responding detenninants of ||Bjj. The determinants of a matrix are, of course, the deter- 
minants of the greatest square matrices contained in it ; similarly, its minor determinants 
of order i are the determinants of the square matrices of the type ixi that are con- 
tained in it. Matrices of the types nx{ni-\-n) and mx{'>n-\-n) are said to be of com- 
plementary types ; if j|A|| and |jBl| be two such matrices, we shall employ the equation 
ia'='b: 
to express that each determinant of |A|| is equal to that determinant of |jBi|, by which it 
A 
is multiplied in the development of the determinant of the square matrix 
B 
m and n are both uneven numbers, the signs of the determinants 
and 
V'hen 
are dif- 
ferent : this occasions a certain ambiguity of sign in the interpretation of the equation 
jA’=:|B|, which, however, will occasion no inconvenience. If the matrices ||A|| and 
I Bjj are at once of the same, and of complementary types ; so that, in this case, the 
equation |A]=|B| may stand for either of two very different sets of equations; but this 
MDCCCLXI. 2 S 
