Solving certain linear equations with coefficients in $\{1,2\}$ without Gaussian elimination.

Question

I'm trying to solve this system of linear equations for every $n\in \mathbb{N}$ ($2$ on the main diagonal, $1$ anywhere else):

$$2x_{1}+x_{2}+...+x_n=1$$

$$x_{1}+2x_{2}+...+x_n=1$$$$...$$

$$x_{1}+x_{2}+...+2x_n=1$$

Obiously I can use Gaussian Elimination, but it seems like a lot of technical work and I don't know how can it be generelized to every $n$.

Any ideas?

Thanks!

Answer 1

By symmetry, all the $x_i$ are equal; so each is equal to $1/(n+1)$.

Answer 2

If you write the system in matrix form $AX=B$, note that the determinant of the matrix $A$ is different from zero. This is easy to prove by induction and developing the determinant along a row or column.

The determinant different from zero means there is a unique solution. Try putting all $x_i$ equal to each other and you'll find the solution.

Answer 3

Hint. Use Guassian elimination for the case when $n=2$. Then do it for $n=3$. Then prove the hypothesis you develop.

Answer 4

You could try to show that there is a unique solution like Raskolnikov suggested. Then, there will be an $a$ such that $$x_1 + x_2 + \ldots + x_n = a$$ All the equations can then be written on the form $$a + x_i = 1$$ which means that all $x_i$ are equal.