Linear diophantine equation of n variables.

Question

I know how to solve a linear Diophantine equation of 2, 3 variables. But is there a way to solve directly a linear Diophantine equation of n variables. For example using matrix?

Answer 1

I guess you'll interested in the following article Linear Diophantine Equations written by William J.Gilbert. It contains the following theorem.

Theorem. To solve the system of linear Diophantine equations $AX=B$, unimodular row reduct $[A^t |I]$ to $[R|T]$, where $R$ is in row-echelon form. Then the system $AX=B$ has a integer solutions if and only if the system $R^tK=B$ has integer solution for $K$, and all the solutions of $AX=B$ are of the form $X=T^tK$.

It also contains one example and one exercise with $3$ variables. So you can compare this with the method you know.