<p>If \(A^2 - A + I = 0\), then the inverse of <i>A</i> is</p>
Step-by-Step Solution
Key Concept: If A² - A + I = 0, rearrange to I = A - A² = A(I - A), which directly reveals A⁻¹ by isolating the identity matrix on one side.
<p><strong>Step 1:</strong> Start with the given equation: A² - A + I = 0</p><p><strong>Step 2:</strong> Rearrange to isolate I: I = A - A² = A(I - A)</p><p><strong>Step 3:</strong> This can be rewritten as: I = A(I - A), which means A · (I - A) = I</p><p><strong>Step 4:</strong> By the definition of matrix inverse, if A · B = I, then B = A⁻¹</p><p><strong>Step 5:</strong> Therefore, A⁻¹ = I - A</p><p><strong>Verification:</strong> A · (I - A) = A - A² = I ✓ (from the original equation)</p><p>∴ Answer: <strong>A⁻¹ = I - A</strong></p>
Correct Answer: C