<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>\((I + A)^{100} =\)</p>
Step-by-Step Solution
Key Concept: When A² = O (nilpotent matrix of index 2), the binomial expansion of (I + A)^n terminates after two terms because all higher powers of A vanish. Use the binomial theorem: (I + A)^100 = I + 100A + C(100,2)A² + ... = I + 100A.
<p><strong>Step 1:</strong> Use the binomial theorem for matrix expansion:</p><p>(I + A)^100 = Σ C(100,k) I^(100-k) A^k for k = 0 to 100</p><p><strong>Step 2:</strong> Since A² = O, we have A^k = O for all k ≥ 2</p><p><strong>Step 3:</strong> Therefore, only the first two terms survive:</p><p>(I + A)^100 = C(100,0)I + C(100,1)A + C(100,2)A² + ...</p><p>(I + A)^100 = I + 100A + 0 + 0 + ...</p><p><strong>Step 4:</strong> Simplify to get the final result:</p><p>∴ Answer: <strong>I + 100A</strong></p>
Correct Answer: D