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Solving linear Diophantine Equations with a matrix equation

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Is there a way to solve a diophantine equation of the type ax+by=c with ONLY a matrix equation, i.e without making use of for example Euclidean algorithm or Modular math. I want to set up the equation above as something of the sort AX=B where X is the[x,y] 2x1 matrix and A and B are some matrices that depend on a,b and c.

I can only get as far as to set it up as $$\begin{bmatrix}a & b \end{bmatrix}\begin{bmatrix}x \\y \end{bmatrix}=\begin{bmatrix}c \end{bmatrix}$$ which doesn't really get me far.

I know this can be solved in a much easier fashion but I want to know if I can make use of linear algebra instead.

Jan 19, 2018

#1
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I do not think there is a way.  I cannot see why there would be a way but I could be wrong.

Jan 20, 2018
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This question was inspired by this post https://math.stackexchange.com/questions/20717/how-to-find-solutions-of-linear-diophantine-ax-by-c/20738#20738

I didn't find the explanation that helpful so I was hoping someone could clarify or even put it into a more systematic setup where it can be solved by a matrix inverse.

Quazars  Jan 20, 2018
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Thanks for showing me that Quazars.  I hope someone can help you. :)

Melody  Jan 20, 2018