According to the answer of znt to the previous version, I revise the question as follows:


Is there a real $(n-1)\times n$ matrix $A$ 
such that $A$ is not a full rank matrix and satisfy $a\_{ii}<0$ and $a\_{ij}>0$ for $i \neq j$ and $\sum\_{i} a\_{ij}<0$ for every $j\leq n-1$.


A stronger version:



> 
> Is there a singular $n\times n$ matrix $A$ such that diagonal entries are negative, off diagonal entries are positive and each column sum up to a negative number?
> 
> 
> 



Is this a full rank matrix?
No, to your stronger question. If an $n \times n$ matrix satisfies the condition you specified, it would be a [strictly diagonally dominant matrix](https://en.wikipedia.org/wiki/Diagonally_dominant_matrix#cite_note-3) which is non-singular.


