Let $A$ be a matrix of order $2 imes 2$, whose entries are from the set $\{0,1,2,3,4,5\}$. If the

Question

Let $A$ be a matrix of order $2 	imes 2$, 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

Professor PC Thomas Classes · Answer

(180) : Let $S=\{0,1,2,3,4,5\}$ It is given that the sum of all entries of $A$ is prime number $p$, where $2<p<8$.\ $	herefore$ The sum of all entries may be 3,5 or 7 .\ Case I : If sum of all entries is 3 . Then possible entries are $(0,0,0,3),(0,0,1,2)$ and ( $0,1,1,1$ ).\ So, number of matrices $=4+12+4=20$.\ Case II : If sum of all entries is 5 . Then possible entries are $(0,0,0,5),(0,0,1,4)$, $(0,0,2,3),(0,1,2,2),(0,1,1,3),(1,1,1,2)$ So number of matrices $=4+12+12+12+$ $12+4=56$\ Case III : If sum of all entries is 7 . Then possible entries are $(0,0,0,7),(0,0,1,6)$, $(0,0,3,4),(0,0,2,5),(0,1,1,5)$, $(0,2,1,4),(0,3,2,2),(1,1,2,3),(2,2,2,1)$ So, number of matrices $=4+12+12+12+$ $12+24+12+12+4=104$\ $	herefore$ Total matrices $=20+56+104=180$\