Results 1 to 1 of 1

Math Help - The determinant of the solutions of a linear congruence

  1. #1
    Newbie
    Joined
    Mar 2008
    Posts
    2

    The determinant of the solutions of a linear congruence

    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

     \sum_{i=1}^3 x_i \; \alpha_{ik} \equiv 0 \quad (a_0) \qquad (k = 1, 2, 3)

    possesses 3 solutions, their determinant has the value

    <br />
	H = \frac{a_0^3}{a_1 \; a_2 \; a_3} <br />

    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:

     a_{11} x_1 + a_{12} x_2 + a_{13} x_3 \equiv 0 \mod a_0

     a_{21} x_1 + a_{22} x_2 + a_{23} x_3 \equiv 0 \mod a_0

     a_{31} x_1 + a_{32} x_2 + a_{33} x_3 \equiv 0 \mod a_0



    (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

     a_1 = \gcd(a_0,e_1),

     a_2 = \gcd(a_0,e_2),

     a_3 = \gcd(a_0,e_3),

    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.
    Last edited by perwiradua; March 3rd 2008 at 08:16 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. determinant of T linear operator
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 27th 2011, 02:28 AM
  2. Solutions in a congruence class
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: January 31st 2009, 05:51 PM
  3. Non linear map - determinant
    Posted in the Advanced Algebra Forum
    Replies: 13
    Last Post: January 17th 2009, 05:26 AM
  4. Determinant and Linear Dependence
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 25th 2008, 04:15 PM
  5. find all solutions to the congruence
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: June 18th 2008, 10:30 AM

Search Tags


/mathhelpforum @mathhelpforum