Hi,(adsbygoogle = window.adsbygoogle || []).push({});

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

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

[tex] \sum_{i=1}^3 x_i \; \alpha_{ik} \equiv 0 \quad (a_{00}) \qquad (k = 1, 2, 3) [/tex]

possesses 3 solutions, their determinant has the value

[tex] H = \frac{a_{00}^3}{a_1 a_2 a_3} [/tex]

where [tex] a_1, a_2, a_3 [/tex] are the greatest common divisor the module(number?) [tex] a_{00} [/tex]

with the elementary divisors [tex] e_1, e_2, e_3 [/tex] respectively of the system [tex] \alpha [/tex].''

The matrix [tex] \alpha [/tex] is a [tex] 3 \times 3 [/tex] matrix with integer entries. My understanding is

that the linear congruence:

[tex] a_{11} x_1 + a_{12} x_2 + a_{13} x_3 \equiv 0 \pmod{a_{00}} [/tex]

[tex] a_{21} x_1 + a_{22} x_2 + a_{23} x_3 \equiv 0 \pmod{a_{00}} [/tex]

[tex] a_{31} x_1 + a_{32} x_2 + a_{33} x_3 \equiv 0 \pmod a_{00}} [/tex]

(where [tex] a_{ij} [/tex] are entries of [tex] \alpha [/tex])

has three incongruent solutions (please prove it), and upon forming a 3x3 matrix

of the solutions the determinant of the matrix is

[tex] \frac{a_{00}^3}{a_1 a_2 a_3} [/tex]

where

[tex] a_1 = \gcd(a_{00},e_1), [/tex]

[tex] a_2 = \gcd(a_{00},e_2), [/tex]

[tex] a_3 = \gcd(a_{00},e_3), [/tex]

where

[tex] e_1, e_2, e_3 [/tex]

are the elementary divisors of the matrix [tex] \alpha [/tex].

I would appreciate it if

(1) a proof of the statement is found,

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

Thank you.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Proof required: The determinant of the solutions of a linear homogeneous congruence

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**