System of Equations-Linear Algebra

Question

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

1. a unique solution at $x=J_n$ where $J_n$ denotes an n-dimensional vector of all 1.
2. No Solution
3. Infinitely many solutions
4. 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?

Answer 1

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.