<p>If \(A^2 - A + I = 0\), then the inverse of \(A\) is</p>
Step-by-Step Solution
Key Concept: From the equation A² - A + I = 0, rearrange to isolate I and factor out A to find A⁻¹ = I - A directly.
<p><strong>Step 1:</strong> Start with the given equation: A² - A + I = 0</p><p><strong>Step 2:</strong> Rearrange to isolate I: A² - A = -I</p><p><strong>Step 3:</strong> Factor out A from the left side: A(A - I) = -I</p><p><strong>Step 4:</strong> Multiply both sides by -1: A(I - A) = I</p><p><strong>Step 5:</strong> Compare with AA⁻¹ = I. This shows that A⁻¹ = I - A</p><p><strong>Verification:</strong> We can verify: A(I - A) = A - A² = A - (A - I) = I ✓</p><p>∴ Answer: <strong>A⁻¹ = I - A</strong></p>
Correct Answer: D