<s>[INST] Proving a system of n linear equations has only one solution I have been given the system:
$-x_1 + 2x_2 + ... + 2x_{n-1} + 2x_n = 1$
$2x_1 - x_2 + ... 2x_{n-1} + 2x_n = 2$
$.$
$.$
$.$
$2x_1 + 2x_2 + ... + 2x_{n-1} - x_n = n$
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?
 [/INST] 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 $n$.
</s>