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Let A be a real 2 × 2 matrix and let I denote the 2 × 2 identity matrix. Also, tr(A) means the sum of the diagonal entries of A. Suppose that A² = I. Statement 1: If A is neither I nor −I, then det(A) = −1. Statement 2: If A is neither I nor −I, then tr(A) ≠ 0.

1. Statement 1 is false, Statement 2 is true
2. Statement 1 is true, Statement 2 is true; moreover, Statement 2 correctly explains Statement 1
3. Statement 1 is true, Statement 2 is true; however, Statement 2 does not correctly explain Statement 1
4. Statement 1 is true, Statement 2 is false

Correct answer: Statement 1 is true, Statement 2 is false

Solution

Statement 1 is true because if A² = I and A is neither I nor -I, it must have eigenvalues of 1 and -1, leading to a determinant of -1. Statement 2 is false because the trace can be zero if the eigenvalues are 1 and -1, which contradicts the claim.

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