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Let A be a square matrix of order 3 such that |A| = 3. If det{adj(A · adj(2A))} = 2^a × 3^b, find the value of a + b.
- 16
- 18
- 20
- 24
Correct answer: 18
Solution
adj(2A) = 2^(3-1) · adj(A) = 4 adj(A). So A · adj(2A) = 4 A · adj(A) = 4|A|I = 12I. Then det(adj(12I)) = |12I|^(3-1) = (12³)² = 12⁶ = (2² · 3)⁶ = 2¹² · 3⁶. Hence a = 12, b = 6, a + b = 18.
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