IXDETEEMIXATE EQUATIONS AND CONGEUENCES. 
295 
Let ||Aj| be a given prime matrix of the type |jK|j a given matrix of the 
same type connected with |jA|j by the equation 
IKj^A-xIAI, (1.) 
which implies that Jc is the greatest divisor of j|K|| ; then the symbolic equation 
liK!|=||/l:||x||A!|, (2.) 
in which jj/^|[ denotes a square matrix of the type n X n, will admit of one solution, and 
one only. 
For, to determine i, ... Av, „, the constituents of the rth horizontal row of p||, we 
have the redundant system 
Kr, i — i ^r, l + ^^2, t 2+ • • • + ,■ ni 
2 * = !, 2, 3, .. . n-\-m 
which is involved in the symbolic equation (2.). The matrix of this system is the prime 
matrix [|Aj' ; and the determinants of its augmented matrix are all equal to zero ; for, by 
virtue of equation (1.), they are equal to the determinants 
R-r, 1) R-r, 2i • • 
Rj, 15 Rl, 25 • • 
* • ^^2, n-k-m 
R’, 15 R 2 , 25 • • 
• • 1^2, n + m 
15 25 • • 
* ' 7i+77» 
in which two horizontal rows are identical. Thus the system (3.), and consequently the 
equation (2.), admits of one solution, and one only. It is evident that the determinant 
of is L The case in which m=0 is not included in this demonstration; its proof, 
however, presents no difficulty, and may be omitted here. 
A particular case of this theorem (that in which n—2) occurs in the ‘ Disquisitiones 
Arithmeticse ’ (see art. 234 of that work). 
Art. 3. If every determinant of the augmented matrix of a redundant system of linear 
congruences be divisible by the modulus, while the greatest divisor of the unaugmented 
matrix is prime to the modulus, the system is resoluble and admits of only one solu- 
tion. For if the modulus be represented by P x Q X R . . ., P, Q, R. . denoting powers of 
unequal piimes, one (at least) of the determinants of the unaugmented matrix is prime 
to P, one (at least) is prime to Q, &c. ; whence it may be inferred that the system is 
resoluble for each of the modules P, Q, R.. ., and admits of only one solution for each 
of them; it is therefore resoluble for their product PxQxR •••, and admits of only 
one solution for that modulus. 
Let ||K|] denote (as in the preceding article) a given matrix of the type nX{n-\-m), of 
which Jc is the greatest divisor; and let it be required to find the complete solution of 
the symbolic equation 
l|K|=Wxl!A||, . (4.) 
in which |jF|j is a square matrix of which the determinant is Jc, ||A|| a prime matrix of 
2 s 2 
