Let A be a matrix of order $2\times 2$, whose entries are from the set $\{0,1,2,3,4,5\}$. If the sum of all the entries of $\mathrm{A}$ is a prime number $\mathrm{p},2<\mathrm{p}<8$, then the number of such matrices $\mathrm{A}$ is :

 $\because$ Sum of all entries of matrix $A$ must be prime $p$ such that $2<p<8$ then sum of entries may be 3,5 or 7 .

If sum is 3 then possible entries are $(0,0,0,3)$, $(0,0,1,2)$ or $(0,1,1,1)$.
$\therefore$ Total number of matrices $=4+4+12=20$
If sum of 5 then possible entries are $(0,0,0,5),(0,0,1,4),(0,0,2,3),(0,1,1,3)$, $(0,1,2,2)$ and $(1,1,1,2)$.
$\therefore$ Total number of matrices $=4+12+12+12+$ $12+4=56$
If sum is 7 then possible entries are $(0,0,2,5),(0,0,3,4),(0,1,1,5),(0,3,3,1)$, $(0,2,2,3),(1,1,1,4),(1,2,2,2),(1,1,2,3)$ and $(0,1,2,4)$
Total number of matrices with sum $7=104$
$\therefore$ Total number of required matrices

$$\begin{array}{rl} & =20+56+104\\ & =180\end{array}$$