<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>\(A^2 - (a+d)A + (ad - bc)I\) is equal to</p>
Step-by-Step Solution
Key Concept: When A² = O (nilpotent matrix), use the Cayley-Hamilton theorem: A satisfies its characteristic equation λ² - (a+d)λ + (ad-bc) = 0, which becomes A² - (a+d)A + (ad-bc)I = O.
<p><strong>Step 1:</strong> For a 2×2 matrix A with trace (a+d) and determinant (ad-bc), the characteristic polynomial is:</p><p>p(λ) = λ² - (trace)λ + (determinant) = λ² - (a+d)λ + (ad-bc)</p><p><strong>Step 2:</strong> By Cayley-Hamilton Theorem, every matrix satisfies its own characteristic equation:</p><p>p(A) = A² - (a+d)A + (ad-bc)I = O</p><p><strong>Step 3:</strong> The condition A² = O confirms A is nilpotent. Substituting into our expression:</p><p>A² - (a+d)A + (ad-bc)I = O - (a+d)A + (ad-bc)I = (ad-bc)I - (a+d)A</p><p>By Cayley-Hamilton, this equals O (the zero matrix).</p><p>∴ Answer: O (Zero Matrix)</p>
Correct Answer: B