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Consider the matrix M = [0 1 a; 1 2 3; 3 b 1] and its adjugate adj M = [-1 1 -1; 8 -6 2; -5 3 -1], where a and b are real numbers. Which of the following statements is/are true?

1. If Mβ = [α; γ; 1], then α - β + γ = 3
2. The sum a + b equals 3
3. The expression adj M⁻¹ + adj M equals -M
4. The determinant of (adj M)² is 81

Correct answer: The sum a + b equals 3

Solution

Matching cofactors of M to the given adjugate forces a=2 and b=1, so det M = -2 and a+b = 3 (option 1 is correct). The stored choice (option 3) claims det((adj M)^2)=81, but det((adj M)^2)=(det M)^4=(-2)^4=16, so it is false.

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