<p>If \(B\) is a \(3 \times 3\) matrix such that \(B^2 = 0\), then \(\det[(I+B)^{50} - 50B]\) is equal to</p>
Step-by-Step Solution
Key Concept: Since B² = 0 (nilpotent matrix), (I+B)ⁿ terminates at the binomial expansion after just two terms: (I+B)ⁿ = I + nB. Therefore (I+B)⁵⁰ = I + 50B, making the determinant calculation trivial.
<p><strong>Step 1:</strong> Recognize that B is nilpotent with B² = 0.</p><p><strong>Step 2:</strong> Expand (I+B)⁵⁰ using the binomial theorem: (I+B)⁵⁰ = I + 50B + C(50,2)B² + ... Since B² = 0, all terms with B² and higher vanish.</p><p><strong>Step 3:</strong> Therefore (I+B)⁵⁰ = I + 50B.</p><p><strong>Step 4:</strong> Compute (I+B)⁵⁰ - 50B = (I + 50B) - 50B = I.</p><p><strong>Step 5:</strong> Calculate det(I) = 1.</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: A