The number of matrices $A={\left[a_{ij}\right]}_{3\times 3}$ with entries in the set $\{1,2,\mathrm{...},9\}$ such that $i+j\le k+l\Rightarrow a_{ij}\le a_{kl}\forall i,j,k,l\in \{1,2,3\}$ is

$$\begin{gathered} i+j=k+l\Rightarrow a_{ij}=a_{kl} \\ i+j\le k+l\Rightarrow a_{ij}\le a_{kl} \end{gathered}$$

So we have to choose 5 numbers in increasing order from $\{1,2,\mathrm{...},9\}$
So no. of matrices $={}^{9+4}C_5={}^{13}C_5$