System of Equations-Linear Algebra Below is a question from a competitive exam
Let $c_1,....c_n$ be scalars, not all zero, such that $\sum_{i=1}^{n}c_ia_i=0$
where $a_i$ are the column vectors in $R^n$.
Consider the set of linear equations
$Ax=b$
where $A=[a_1....a_n]$ and $b=\sum_{i=1}^{n}a_i$. The set of equations has


*

*a unique solution at $x=J_n$ where $J_n$ denotes an n-dimensional vector of all 1.

*No Solution

*Infinitely many solutions

*Finitely many solutions.


**My Attempt :*
  From given information, it is clear that the columns of the matrix A are dependent and hence there exists a non-zero vector $x_n$ where it denotes solution to the null space, such that $Ax_n=0$...(a)
Now, it is also given that $b=\sum_{i=1}^{n}a_i$, which means a column vector of all 1,say $x_p$ a particular solution to this b, is such that $Ax_p=b$ ..(b)
From (a) and (b),if c is a constant,then $A(x_p+c.x_n)=b$ and hence this system seems to have infinite solutions.
Is my attempt correct?

Yes, your attempt is correct. 
There are infinitely many solutions. The matrix $A$ is singular and we have illustrated a particular solution, hence there must be infinitely many solutions.
