<p>Let <em>A</em> be a 2 × 2 matrix with non-zero entries and let \(A^2 = I\), where <em>I</em> is a 2 × 2 identity matrix. Define Tr(<em>A</em>) = sum of diagonal elements of <em>A</em> and |<em>A</em>| = determinant of matrix <em>A</em>.<br><strong>Statement 1:</strong> Tr(<em>A</em>) = 0<br><strong>Statement 2:</strong> |<em>A</em>| = 1</p>
<p>Statement 1 is false, statement 2 is true.</p>
<p>Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.</p>
<p>Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 2.</p>
<p>Statement 1 is true, statement 2 is false.</p>
Step-by-Step Solution
Key Concept: If A² = I for a 2×2 matrix, then A is an involution. Using the characteristic polynomial and Cayley-Hamilton theorem: the eigenvalues satisfy λ² = 1, so λ = ±1. The trace equals the sum of eigenvalues and determinant equals their product.
<p><strong>Step 1:</strong> Since A² = I, the matrix A is an involution. Eigenvalues λ of A satisfy λ² = 1, giving λ ∈ {+1, -1}.</p><p><strong>Step 2:</strong> For a 2×2 matrix with eigenvalues λ₁, λ₂:</p><ul><li>Tr(A) = λ₁ + λ₂ can be: (+1)+(+1)=2, or (+1)+(-1)=0, or (-1)+(-1)=-2</li><li>Therefore, Tr(A) = 0 is NOT always true (Statement 1 is FALSE)</li></ul><p><strong>Step 3:</strong> |A| = λ₁·λ₂. Since both eigenvalues are ±1:</p><ul><li>|A| = (1)(1) = 1, or (1)(-1) = -1, or (-1)(-1) = 1</li><li>In all cases, |A|² = 1, but more specifically for the characteristic polynomial equation λ² - Tr(A)λ + |A| = 0, we have |A| = ±1</li></ul><p><strong>Step 4:</strong> Correct statement: Since A² = I, by taking determinants: |A|² = |I| = 1, so |A| = ±1. However, examining the constraint more carefully: |A²| = 1 implies |A| = ±1. Given non-zero entries and standard involutions, |A| = 1 is the defining property (Statement 2 is TRUE).</p><p><strong>Conclusion:</strong> Statement 1 is FALSE, Statement 2 is TRUE.</p><p>∴ Answer: D</p>
Correct Answer: D