294 
ME. H. J. S. SMITH ON SYSTEMS OF LINEAE 
also is an imperfection of the notation here employed, which it is sufficient to have 
pointed out. If A' denote any quantity Avhatever, it is hardly necessary to state that the 
equality 
|Al=A:x|Bl 
implies that the determinants of ||A|| are respectively k times the corresponding determi- 
nants of ||B||. 
Let llPjl be a square matrix of the type n'X.n^ and ||Qj| a matrix of the type nx{n-\-m) 
(where we shall understand by the matrix compounded of ||P|| and |]Qj|, the matrix 
I X|| of the same type as ||Q||, the constituents of which are defined by the equation 
and we shall write 
Pi,) Qi^y + Pj^a Qa.yd" 
||XI|=||P||xl|Q||; 
in this equation |jQ|| is said to be pre multiplied by ||Pj|, and ||P|| to be post-multiplied by 
[Q]. This definition will suffice for our present purpose ; as the only case of composi- 
tion which we shall have to consider, is that in which the vertical dimensions of the 
matrices to be compounded are all equal, and in which every premultiplying matrix is 
square, so that if an oblong matrix present itself at all in a series of matrices to be com- 
pounded, it will occupy the last place in the series. 
By the greatest divisor of a matrix we are to understand the greatest common divisor 
of the determinants of the matrix. If the matrix be square, its greatest divisor is, con- 
sequently, the determinant of the matrix. A. prime matrix is one of which the greatest 
divisor is unity ; i. e. the determinants of which are relatively prime. A prime square 
matrix {i. e. a matrix of which the determinant is unity) we shall call a unit-matrix. 
In any system of linear equations, whether defective or redundant, or neither, we 
shall understand by the matrix of the system the matrix formed by the coefficients of 
the unknown quantities. If to this matrix we add an additional vertical column, com- 
posed of the absolute terms of the equations, the resulting matrix we shall term (for 
brevity) the augmented matrix of the system. 
Lastly, when we have occasion to consider square matrices, the constituents of which, 
excepting those on the principal diameter, are zero, we shall represent them by symbols 
of the form 
where rp, p., ... g^n fhe constituents of the principal diameter. 
Art. 2. If every determinant of the augmented matrix of a redundant system of linear 
equations is equal to zero, while the determinants of the unaugmented matrix are not 
all equal to zero, the system admits of one solution, and one only. And in particular if 
the matrix of the system be a prime matrix, the values of the unknown quantities which 
satisfy the system are integral numbers. For these values may be expressed as fractions 
haring for their denominators any one of the determinants of the matrix; and these 
determinants are relatively prime. 
