<p>Let \(f(x) = \dfrac{1+x}{1-x}\). If <i>A</i> is a matrix for which \(A^3 = O\), then \(f(A)\) is</p>
Step-by-Step Solution
Key Concept: When A³ = O (nilpotent matrix), use the functional definition f(A) = (I + A)(I - A)⁻¹ and expand (I - A)⁻¹ as a geometric series: (I - A)⁻¹ = I + A + A² (since A³ = O terminates the series).
<p><strong>Step 1:</strong> Given f(x) = (1+x)/(1-x), we need f(A) = (I + A)(I - A)⁻¹</p><p><strong>Step 2:</strong> Since A³ = O, find (I - A)⁻¹ using the geometric series expansion. For |A| < 1 (or nilpotent A):<br/>∑(A)ⁿ = I + A + A² + A³ + ... = I + A + A² (since A³ = O)</p><p><strong>Step 3:</strong> Verify: (I - A)(I + A + A²) = I + A + A² - A - A² - A³ = I - O = I ✓<br/>Therefore (I - A)⁻¹ = I + A + A²</p><p><strong>Step 4:</strong> Compute f(A):<br/>f(A) = (I + A)(I + A + A²)<br/>= I + A + A² + A + A² + A³<br/>= I + 2A + 2A² + O<br/>= I + 2A + 2A²</p><p><strong>Step 5:</strong> This can be written as f(A) = I + 2(A + A²) or factored based on answer choices.</p><p>∴ Answer: B</p>
Correct Answer: B