<p>If \(c_{22}c_{33} - c_{23}c_{32} = \det(A^{20}) = 2^{20} \equiv 2^m\), find the value of \(m\).</p>
Step-by-Step Solution
Key Concept: The expression c₂₂c₃₃ - c₂₃c₃₂ represents the (1,1) cofactor expansion element of a 3×3 matrix. Recognizing that det(A²⁰) = 2²⁰ and matching it with 2^m directly gives m = 20.
<p><strong>Step 1:</strong> Recognize that c₂₂c₃₃ - c₂₃c₃₂ is the (1,1) minor determinant of the cofactor matrix, but the problem statement directly provides that det(A²⁰) = 2²⁰.</p><p><strong>Step 2:</strong> The condition states det(A²⁰) ≡ 2^m, meaning 2²⁰ ≡ 2^m.</p><p><strong>Step 3:</strong> Comparing exponents: 2²⁰ = 2^m implies m = 20.</p><p>∴ Answer: m = <strong>20</strong></p>
Correct Answer: 20