<p>If \(A^2 - A + I = 0\) where \(A\) is a square matrix and \(I\) is the unit matrix of the same order then \(A^{-1}\) is</p>
Step-by-Step Solution
Key Concept: From the equation A² - A + I = 0, rearrange to isolate a term containing the identity matrix, then factor out A to reveal A⁻¹ as a linear combination of A and I.
<p><strong>Step 1:</strong> Given: A² - A + I = 0</p><p><strong>Step 2:</strong> Rearrange the equation: A² - A + I = 0 → 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> By definition of inverse matrix, since A(I - A) = I, we have A⁻¹ = I - A</p><p><strong>Verification:</strong> We can verify: A · A⁻¹ = A(I - A) = A - A² = A - (A - I) = I ✓</p><p>∴ <strong>Answer: A⁻¹ = I - A</strong></p>
Correct Answer: C