<p>If \(\text{adj } B = A\), \(|P| = |Q| = 1\), then \(\text{adj}(Q^{-1} B P^{-1})\) is</p>
Step-by-Step Solution
Key Concept: Use the property that adj(AB) = (adj B)(adj A) and adj(A⁻¹) = (adj A)⁻¹/|A|, combined with the fact that adj B = A and |P| = |Q| = 1 to simplify the adjugate of a product.
<p><strong>Step 1:</strong> Start with adj(Q⁻¹BP⁻¹). Use the property that for matrices with determinant 1, adj(A⁻¹) = (adj A)⁻¹.</p><p><strong>Step 2:</strong> Apply adj(ABC) = (adj C)(adj B)(adj A) in reverse order: adj(Q⁻¹BP⁻¹) = adj(P⁻¹) · adj(B) · adj(Q⁻¹).</p><p><strong>Step 3:</strong> Since |P| = |Q| = 1, we have adj(P⁻¹) = (adj P)⁻¹ and adj(Q⁻¹) = (adj Q)⁻¹. But more directly, adj(P⁻¹) = P and adj(Q⁻¹) = Q (since for |M| = 1, M · adj M = I implies adj(M⁻¹) = M).</p><p><strong>Step 4:</strong> Therefore: adj(Q⁻¹BP⁻¹) = P · adj(B) · Q = P · A · Q.</p><p>∴ Answer: C</p>
Correct Answer: C