<p><strong>For Problems 1–3</strong><br>Let \(A\) be a matrix of order \(2 \times 2\) such that \(A^2 = O\).<br><br>\(\text{tr}(A)\) is equal to</p>
Step-by-Step Solution
Key Concept: If A² = O for a 2×2 matrix, then all eigenvalues of A must be 0. By the trace-eigenvalue relationship, tr(A) equals the sum of eigenvalues, which must be 0.
<p><strong>Step 1:</strong> For a 2×2 matrix A with A² = O, find the eigenvalues. If λ is an eigenvalue of A, then λ² is an eigenvalue of A². Since A² = O, we have λ² = 0, so λ = 0.</p><p><strong>Step 2:</strong> Both eigenvalues of A are 0 (the only eigenvalue is 0 with algebraic multiplicity 2).</p><p><strong>Step 3:</strong> By the fundamental property of trace, tr(A) = sum of all eigenvalues = 0 + 0 = 0.</p><p><strong>Step 4:</strong> Alternatively, if A = [a, b; c, d], then tr(A) = a + d. The condition A² = O gives us constraints that force a + d = 0.</p><p>∴ Answer: tr(A) = 0 (Option B)</p>
Correct Answer: B