Using maximum flow algorithm to check existence of a matrix

Question

Using the maximum flow algorithm, I have to determine if there exists a $3\times 3$ matrix $P$ (such that all elements are $\geq 0$). I'm given:

• The maximum values of the row sums
• The column sums
• The maximum value each element can take

I have a pretty good idea of how to build the network:

• Have three vertices $r_1, r_2, r_3$ represent the rows and have another three vertices $c_1, c_2, c_3$ represent the columns, with the capacity of each arc from $r_i$ to $c_j$ representing the maximum value of $i,j$-th entry in $P$
• Have a source $s$ with arcs between $s$ and each $r_i$ representing the maximum possible value for the sum of that row
• Have a sink $t$ with edges between each $c_j$ and $t$ representing the sum of that column

However, I'm at a loss on how to use the algorithm to check if such a $P$ can actually exist.