Matrices & Determinants
Inverse of a Matrix
Grade None
Question:
<p>Given <em>A</em><sup>2</sup> − <em>A</em> + <em>I</em> = <em>O</em>, then <em>A</em><sup>−1</sup> equals:</p>
<p>(1) \(A + I\)</p>
<p>(2) \(A - I\)</p>
<p>(3) \(I - A\)</p>
<p>(4) \(I - A\) (i.e., \(A^{-1} = I - A\))</p>
Step-by-Step Solution
Key Concept: If A² - A + I = O, rearrange to isolate A, then multiply both sides by A⁻¹ to express the inverse in terms of A itself.
<p><strong>Step 1:</strong> Start with the given equation: A² - A + I = O</p><p><strong>Step 2:</strong> Rearrange: A² - A + I = O → A² + I = A</p><p><strong>Step 3:</strong> Multiply both sides by A⁻¹ from the right: (A² - A + I)A⁻¹ = OA⁻¹</p><p><strong>Step 4:</strong> Expand: A·A·A⁻¹ - A·A⁻¹ + I·A⁻¹ = O</p><p><strong>Step 5:</strong> Simplify: A·I - I + A⁻¹ = O → A - I + A⁻¹ = O</p><p><strong>Step 6:</strong> Solve for A⁻¹: A⁻¹ = I - A</p><p>∴ Answer: D (A⁻¹ = I - A)</p>
Correct Answer: D