<p>If <em>P</em> is an orthogonal matrix and \(Q = PAP^T\) and \(x = P^T Q^{1000} P\), then \(x^{-1}\) is, where <em>A</em> is involutary matrix</p>
Step-by-Step Solution
Key Concept: Since P is orthogonal (P^T P = I) and A is involutary (A² = I), the conjugation Q = PAP^T preserves the involutary property. This means Q² = I, so Q^1000 = I, making x = P^T·I·P = I, hence x^(-1) = I.
<p><strong>Step 1:</strong> Since A is involutary, A² = I (given property)</p><p><strong>Step 2:</strong> Find Q²: Q² = (PAP^T)(PAP^T) = PA(P^T P)AP^T = PA·I·AP^T = PA²P^T = P·I·P^T = PP^T = I</p><p><strong>Step 3:</strong> Since Q² = I, Q is also involutary. Therefore Q^(1000) = (Q²)^500 = I^500 = I</p><p><strong>Step 4:</strong> Calculate x: x = P^T Q^(1000) P = P^T·I·P = P^T P = I (since P is orthogonal)</p><p><strong>Step 5:</strong> Find x^(-1): Since x = I, we have x^(-1) = I^(-1) = I</p><p>∴ Answer: <strong>x^(-1) = I</strong></p>
Correct Answer: A