Let $\mathrm{A}=\left[\mathrm{a}_{i j}\right], \mathrm{a}_{i j} \in \mathbb{Z} \cap[0,4], 1 \leq i, j \leq 2$.

The number of matrices A such that the sum of all entries is a prime number $\mathrm{p} \in(2,13)$ is __________.

$A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{13} & a_{14}\end{array}\right]$ <br/><br/>Such that $\Sigma a_{i i}=3,5,7$ or 11 <br/><br/>Then for sum 3, the possible entries are $(0,0,0,3)$, $(0,0,1,2),(0,1,1,1)$. <br/><br/>Then total number of possible matrices $=4+12+4$ $=20$ <br/><br/>For sum 5 the possible entries are $(0,0,1,4)$, $(0,0,2,3),(0,1,2,2),(0,1,1,3)$ and $(1,1,1,2)$. <br/><br/>$\therefore $ Total possible matrices $=12+12+12+12+4=52$ <br/><br/>For sum 7 the possible entries are $(0,0,3,4)$, $(0,2,2,3),(0,1,2,4),(0,1,3,3),(1,2,2,2)$, $(1,1,2,3)$ and $(1,1,1,4)$. <br/><br/>$\therefore $ Total possible matrices $=80$ <br/><br/>For sum 11 the possible entries are $(0,3,4,4)$, $(1,2,4,4),(2,3,3,3),(2,2,3,4)$. <br/><br/>$\therefore $ Total number of matrices $=52$ <br/><br/>$\therefore $ Total required matrices $=20+52+80+52$ $=204$