Question

Let A be a matrix of order 2 × 2, whose entries are from the set {0, 1, 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

$∵$ 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 $(0, 1, 1, 1).$
$∴$ Total number of matrices $= 4+4+12=20$
If sum of 5 then possible entries are
$(0, 0, 0, 5), (0, 0, 1, 4), (0, 0, 2, 3), (0, 1, 1, 3), (0, 1, 2, 2) \ and\ (1, 1, 1, 2).$
$∴$ 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 $7=104$
$∴$ Total number of required matrices$= 20+56+104=180$