# Thread: could anyone tell me how to find all solutions of linear Cramer equation system

1. ## could anyone tell me how to find all solutions of linear Cramer equation system

Hello.

Does anyone know how to tind all solutions to following equation system where umbers a(ij) are integers.

I know how to use gaussian ellimination method, but I still dont know how to write down all solutions.

2. ## Re: could anyone tell me how to find all solutions of linear Cramer equation system

Hey rain1.

First of all you should collect all the x_i terms to the LHS and then reduce that matrix.

What did you get for the reduced matrix? Is it full rank or did you get some rows of zeroes when you reduced it?

3. ## Re: could anyone tell me how to find all solutions of linear Cramer equation system

$\displaystyle (2a_{11}-1)x_1 + 2a_{12}x_2 + 2a_{13}x_3 + ...+ 2a_{1n}x_n = 0$
$\displaystyle 2a_{21}x_1 + (2a_{22}-1)x_2 + 2a_{23}x_3 + ...+ 2a_{2n}x_n = 0$
..
$\displaystyle 2a_{n1}x_1 + 2a_{n2}x_2 + 2a_{n3}x_3 + ...+ (2a_{nn}-1)x_n =0$

4. ## Re: could anyone tell me how to find all solutions of linear Cramer equation system

Since those are all homogenous linear equations, except for special circumstances, the only solution will be all $\displaystyle a_{ij}= 0$. The "special circumstances" will be that 1/2 is an eigenvalue of the original matrix. In that case the will be an infinite number of solutions.

5. ## Re: could anyone tell me how to find all solutions of linear Cramer equation system

But how do I write down that infinate number of solutions?

Edit: What is exactly eigenvalue of the matrix and what do I do with that?

If I want to shot that 0 vector is only solution, then how would I do that?

6. ## Re: could anyone tell me how to find all solutions of linear Cramer equation system

Originally Posted by HallsofIvy
Since those are all homogenous linear equations, except for special circumstances, the only solution will be all $\displaystyle a_{ij}= 0$. The "special circumstances" will be that 1/2 is an eigenvalue of the original matrix. In that case the will be an infinite number of solutions.
The solutions will be the eigen vectors corresponding to the eigen value 1/2. When these are normalised there will be 1 or 2 or ... or n of these depending on the multiplicity of the eigen value.

I would normally use software for this task, but if you must do it by hand start by reducing the homogeneous system to upper triangular form then assign any of the x's corresponding to null rows arbitary values and solve for the other x's in terms of these.

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