Hi,

I came across a statement by Frobenius, in German, and online translator says,

--> begin statement

``Accordingly a system of 3 homogeneous linear congruences with 3 unknowns

possesses 3 solutions, their determinant has the value

where a1, a2, a3 are the greatest common divisor the module(number?) a0 with the elementary divisors e1, e2, e3 respectively of the system \alpha.''

--> end statement

The matrix \alpha is a 3x3 matrix with integer entries. My understanding is

that the linear congruence:

(where aij are entries of \alpha)

has three incongruent solutions (please prove it), and upon forming a 3x3 matrix of the solutions the determinant of the matrix is a0^3/(a1*a2*a3) where

where e1, e2 and e3 are the elementary divisors of the matrix \alpha.

I would appreciate it if

(1) a proof of the statement is found,

(2) an example illustrating the fact of the statement is provided.

in the reply to this post.

Thank you.