Suppose $r_i$, $1 \le i \le n$, and $c_j$, $1 \le j \le m$, are nonnegative integers. When does there exist an $n \times m$ matrix in $\text{Mat}_{n \times m} (\mathbb{Z}^+)$, i.e. nonnegative entries, such that $r_i$ is the sum of the entries in its $i$th row and $c_j$ is the sum of the entries in its $j$th columns?
Combinatorics – Constraints on Sum of Rows and Columns of Matrix
algebra-precalculuscombinatoricsmatrices
Best Answer
Non-negative integer solutions exist whenever the (also necessary) "balance condition" is met:
$$ \sum_i r_i = \sum_j c_j $$
which simply sums the total matrix entries in two ways.
There is a survey paper by Alexander Barvinok on this: http://www.math.lsa.umich.edu/~barvinok/linalg.pdf