$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.


What should $A$ satisfy to guarantee the equation set have only zero solution?



Is this a linear optimization problem? $Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative
If a nonzero $x\geq 0$ is a solution to $Ax=0$ then $\sum\_j x\_j=a>0$, and $x$ is also a solution to the system 
$$A x=0,\quad 0\leq x,\quad e^\top x\leq a,\qquad\qquad\qquad\qquad(\*) $$
where $e$ denotes the all-1 vector.
It is a standard application of Farkas lemma in an appropriate form (one might want to use instead $e^\top x=a$) to check whether $(\*)$ has a solution. Or, indeed, you may maximise $\sum\_j x\_j$ subject to $(\*)$ and see whether it has maximum $a$ or $0$.


