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3x3 matrices are formed using elements from the set {-3, -2, -1, 0, 1, 2, 3}. How many such matrices have a trace (sum of diagonal elements) of at least 7?
- 10 * (7⁷)
- 10 * (7⁶)
- 7⁷
- 7⁶
Correct answer: 10 * (7⁶)
Solution
Off-diagonal elements: 6 entries, each chosen freely from 7 values => 7⁶ ways. Diagonal: need a11+a22+a33 >= 7 where each is in {-3,...,3}. Max = 9, min = -9. Count ordered triples (a,b,c) with sum >= 7: Sum=9: (3,3,3) = 1 way. Sum=8: (3,3,2) permutations = 3 ways. Sum=7: (3,3,1),(3,2,2) permutations = 3+3=6 ways. Total = 1+3+6 = 10. Total matrices = 10 * 7⁶.
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