Question

The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _________.

Answer 1

Correct Answer: 282

To find the number of matrices of order 3×3 with entries either 0 or 1, such that the sum of all the entries is a prime number, we begin by determining the possible sums. The total number of entries in a 3×3 matrix is 9. Each entry can either be 0 or 1, so the sum of the entries can range from 0 to 9. 

Next, identify which numbers within this range are prime. The prime numbers ≤9 are 2, 3, 5, and 7.

The number of matrices with a given sum of entries is calculated by counting the number of ways to achieve the sum \( k \), using combinations from \( 9 \) total positions. The formula to determine this is \( \binom{9}{k} \).

Prime Sum \( k \)	Number of matrices
2	\(\binom{9}{2} = 36\)
3	\(\binom{9}{3} = 84\)
5	\(\binom{9}{5} = 126\)
7	\(\binom{9}{7} = \binom{9}{2} = 36\)

The total number of matrices where the sum of entries is a prime is the sum of the matrices with sum \( k = 2, 3, 5, \) and \( 7 \):

\(36 + 84 + 126 + 36 = 282\)

The calculated total, 282, lies within the expected range of 282,282. Thus, the number of such matrices is 282.