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.