Determinant of an $n\times n$ matrix where $n$ is unknown

Question

I´m trying to solve the following problem:

Let $a$ be a scalar. Consider the set $\left\{ v_{1},v_{2},...,v_{n} \right\} \subseteq \mathbb{R}^{n}$ such that the $i$th element of $v_{i}$ is equal to 1 and the rest of the elements are equal to $a$ ($i=1,2,...,n)$. The scalar $a$ needs to meet 2 conditions for the set $\left\{ v_{1},v_{2},...,v_{n} \right\}$ to be linearly independent. ¿Which are these 2 conditions?

This is a translation I made of the original problem which is in spanish. I think there is a possibility that I´m not translating it well so I´m attaching a picture of the problem at the end of this post if anyone that speaks spanish sees this post can tell me if I´m making a correct translation (I don´t know if this is against the rules so sorry if I break one).

Either way, what I understand is this:$$v_{1}=\begin{pmatrix} 1\\ a\\ a\\ ...\\ a \end{pmatrix}, v_{2}=\begin{pmatrix} a\\ 1\\ a\\ ...\\ a \end{pmatrix},...,v_{n}=\begin{pmatrix} a\\ a\\ a\\ ...\\ 1 \end{pmatrix}$$

So, for the set to be linearly independent: $$M_{n \times n}\overrightarrow{x}=\overrightarrow{0}\to \overrightarrow{x}=\overrightarrow{0}$$ If the determinant of $M$ is $0$, then the system of equations has no unique solution (infinite or none) and therefore the set wouldn´t be L.I. so $|M|\neq0$ . The thing is that I don´t know how to compute the determinant of a $n\times n$ matrix. I founded the determinant of the following matrices: $$\begin{vmatrix} 1 &a &a \\ a &1 &a \\ a &a &1 \end{vmatrix} , \begin{vmatrix} 1 &a &a &a \\ a &1 &a &a \\ a &a &1 &a \\ a &a &a &1 \end{vmatrix}, \begin{vmatrix} 1 & a&a &a &a \\ a& 1& a& a & a\\ a&a &1 & a & a \\ a& a & a&1 &a \\ a& a& a& a&1 \end{vmatrix}$$ and the answers were $(1-a)^{2}(2a+1),(1-a)^{3}(3a+1)$ and $(1-a)^{4}(4a+1)$ respectively. Supposing that $|M|=(1-a)^{n-1}((n-1)a+1)$, by algebraic manipulation, I got that $a\neq1$ and $a\neq1/(1-n)$.

The thing is that i don´t think i can´t just assume that $|M|=(1-a)^{n-1}((n-1)a+1)$ based on "smaller versions" of $M$, or can I? If not, how do I prove it? by induction? Also, are this really the 2 conditions the problem is asking for?

Original problem in spanish: