If $Ax=c(x)e, \forall x$, then $A$ has rank one

Question

How to prove if $Ax=c(x)e, \forall x$, then $A$ has rank one?

1. $e$ is a vector with all entries one.
2. $c(x)\in \mathbb{R}$, which is a constant depending on $x$

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My method is by Gaussian Elimination:

$[A\ \ | \ c(x)e] \rightarrow [A_1\ \ | \ \ $$\begin{bmatrix}c(x) \\0 \\0 \\ \vdots \\ 0\end{bmatrix}$$ \ \ ]$.

So can I say $A$ has rank one by this?

Answer 1

The rank is the dimension of the range, i.e. of the set of all vectors $Ax$. Those vectors are all multiples of $e$, so the rank is at most $1$. It would be $0$ if all $c(x) = 0$, otherwise it is $1$.