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Step-by-Step Solution
Step 1: Identify the possible prime sums
The entries of the 2×2 matrix A are from the set {0,1,2,3,4,5}, and the sum of these four entries must be a prime number p such that 2 < p < 8. The only possibilities for p under these conditions are 3, 5, and 7.
Step 2: Count the number of matrices having sum = 3
We need non-negative integer solutions to the equation:
a + b + c + d = 3 , where a, b, c, d \in \{0,1,2,3,4,5\}
However, each variable cannot exceed 5. The possible combinations that sum to 3 without exceeding 5 are:
(3, 0, 0, 0) and its permutations
(2, 1, 0, 0) and its permutations
(1, 1, 1, 0) and its permutations
Counting permutations:
(3,0,0,0) has 4 permutations (since 3 can occupy any of the 4 spots).
(2,1,0,0) has 12 permutations (the 2 can be placed in 4 ways and the 1 can be placed in 3 of the remaining spots, giving 4×3 = 12 total distinct arrangements).
(1,1,1,0) has 4 permutations (the 0 can occupy any of the 4 spots and the 1’s fill the rest).
So, total number of such matrices for sum = 3 is:
4 + 12 + 4 = 20
Step 3: Count the number of matrices having sum = 5
Now we solve:
a + b + c + d = 5 , where a, b, c, d \in \{0,1,2,3,4,5\}
Again, no entry can exceed 5. Some representative patterns include:
(5, 0, 0, 0) and permutations
(4, 1, 0, 0) and permutations
(3, 2, 0, 0) and permutations
(3, 1, 1, 0) and permutations
(2, 2, 1, 0) and permutations
(2, 1, 1, 1) and permutations
The solution given enumerates these systematically, and the total count is found to be 56.
Step 4: Count the number of matrices having sum = 7
Similarly, for the sum = 7, we consider:
a + b + c + d = 7 , where a, b, c, d \in \{0,1,2,3,4,5\}
The detailed enumeration of all valid combinations (ensuring none of the numbers exceeds 5) is done similarly. The given detailed list yields a total of 104 matrices for sum = 7.
Step 5: Sum up the counts for the three prime sums
Adding all these results together gives the total number of matrices with sum of entries equal to a prime p (3, 5, or 7):
20 + 56 + 104 = 180
Final Answer
The number of 2×2 matrices with entries from {0,1,2,3,4,5} whose sum of entries is a prime between 2 and 8 is:
180
