Let $R:=\mathbb{Z}[X_1,X_2,\dots,X_{mn}]$. Suppose $A=(f_{ij})$ is a $m\times n$ matrix with entries in $R$ such that
(1)there is no zero column in $A$;
(2)for each $i,j$, either $f_{ij}=0$ or $f_{ij}=X_k$ for some $k\in \{1,2,…,mn\}$;
(3)if $f_{ij}\neq 0$ and $f_{i'j'} \neq 0$, then $f_{ij} \neq f_{i'j'}$.
Is it true that if there exists nonzero $f_1,f_2,\dots,f_m$(here $f_i\neq 0$ for every $i$) in $R$ such that $(f_1\ f_2\ \dots\ f_m)A=0$, then we must have $m>n$?