<p>If <i>A</i> is a square matrix such that <i>A</i><sup>2</sup> = <i>A</i>, then det(<i>A</i>) is equal to</p>
Step-by-Step Solution
Key Concept: If A² = A, then A(A−I) = O. Taking determinants and using the fact that eigenvalues satisfy λ² = λ, we get λ ∈ {0,1}.
<p><strong>Step 1:</strong> Given <i>A</i><sup>2</sup> = <i>A</i>, we have <i>A</i><sup>2</sup> − <i>A</i> = <i>O</i>, so <i>A</i>(<i>A</i> − <i>I</i>) = <i>O</i>.</p><p><strong>Step 2:</strong> Taking determinants: det(<i>A</i>) · det(<i>A</i> − <i>I</i>) = 0.</p><p><strong>Step 3:</strong> This means either det(<i>A</i>) = 0 or det(<i>A</i> − <i>I</i>) = 0. If det(<i>A</i> − <i>I</i>) = 0, then 1 is an eigenvalue of <i>A</i>. Since <i>A</i><sup>2</sup> = <i>A</i>, if λ is an eigenvalue, then λ<sup>2</sup> = λ, giving λ = 0 or λ = 1. Therefore det(<i>A</i>) = 0 or 1.</p><p>∴ Answer is (a).</p>
Correct Answer: A