Proving a system of n linear equations has only one solution I have been given the system:
$-x_1 + 2x_2 + ... + 2x_{n-1} + 2x_n = 1$
$2x_1 - x_2 + ... 2x_{n-1} + 2x_n = 2$
$.$
$.$
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$2x_1 + 2x_2 + ... + 2x_{n-1} - x_n = n$
And the assignment to prove that it has only one solution. I am aware of the existence of discriminants and their role of solving system of linear equations, although I have no idea on how to use that to prove this system has only one solution. (And I currently lack materials on infinite determinants and recurrent relations, any links would be appreciated).
Could anyone provide suggestions or hints on how to prove this system has exactly one solution?

Your equation is of the form $Ax = b$ where $$A = \begin{bmatrix}-1 & 2 & 2 & 2 & 2 & \cdots & 2 & 2\\ 2 & - 1 & 2 & 2 & 2 &\cdots & 2 & 2\\ 2 & 2 & -1 & 2 & 2 &\cdots & 2 & 2\\ 2 & 2 & 2 & -1 & 2 &\cdots & 2 & 2\\ 2 & 2 & 2 & 2 & -1 &\cdots & 2 & 2\\ \vdots & \vdots & \vdots & \vdots & \vdots &\ddots & \vdots & \vdots\\ 2 & 2 & 2 & 2 & 2 &\cdots & -1 & 2\\ 2 & 2 & 2 & 2 & 2 &\cdots & 2 & -1 \end{bmatrix}$$
We then have
$$A = -3I_{n \times n} + 2 \begin{bmatrix} 1\\ 1\\ 1\\ 1\\ 1\\ \vdots\\ 1\\ 1\end{bmatrix} \begin{bmatrix} 1& 1& 1& 1& 1& \cdots& 1& 1\end{bmatrix}$$
By Sherman-Morrison, we have
$$A^{-1} = -\dfrac13 I_{n \times n}-\dfrac2{3(3-2n)}\begin{bmatrix} 1\\ 1\\ 1\\ 1\\ 1\\ \vdots\\ 1\\ 1\end{bmatrix}\begin{bmatrix} 1& 1& 1& 1& 1& \cdots& 1& 1\end{bmatrix}$$
Clearly, $A^{-1}$ exists for all $n$.
