Any positive number (i.e., a matrix of size $1\times 1$) $A>0$ can be written as $A=\frac{1}{A}A^2,$ where in the last product one of the factors is necessarily $\geq 1$ and the other one is $\leq 1$.
Let now $A$ be any matrix with positive entries. Is it always possible to find matrices $B$ and $C$ such that $A=BC$ with all entries of $B$ being not less than $1$ and all entries of $C$ being not bigger than $1$?
Yes, this can be done very simply. Let $d$ be a number smaller than $1$ and smaller than the smallest entry of $A$, and let $I$ denote the identity matrix with size equal to the number of columns in $A$. We then have $$ A = (A/d) (d I). $$ All entries of $B = A/d$ are larger than $1$, and all entries of $d I$ are either $0$ or $d$ and are therefore smaller than $1$.