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Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr(A), the sum of diagonal entries of A. Assume that A² = I. Statement -1: If A ≠ I and A ≠ −I, then det A = −1. Statement -2: If A ≠ I and A ≠ −I, then tr(A) ≠ 0. Which one of the following is true? (1) Statement −1 is false, Statement −2 is true (2) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1 (3) Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1 (4) Statement −1 is true, Statement −2 is false

1. (1) Statement −1 is false, Statement −2 is true
2. (2) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1
3. (3) Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1
4. (4) Statement −1 is true, Statement −2 is false

Correct answer: (4) Statement −1 is true, Statement −2 is false

Solution

Statement -1 is true because if A is a 2x2 matrix such that A² = I and A is neither I nor -I, then the determinant must be -1, as the eigenvalues of A would be ±1. However, Statement -2 is false because the trace of A can be zero in this case, specifically when the eigenvalues are 1 and -1.

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