Question

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 $A$ is a prime number $p,2<p<8$, then the number of such matrices $A$ is .

Answer 1

$\because$ 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 $=4+12+4=20$
If sum is $5,$ 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 $(1,1,1,2)$.
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 $=12+12+12+12+12+4+4+12+24=104$
$\therefore$ Total number of required matrices
$=20+56+104$
$=180$