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Let A and B be two invertible matrices of order 3 × 3. If det(ABA^T) = 8 and det(AB⁻¹) = 8, then det(BA⁻¹B^T) is equal to:

1. 1/4
2. 16
3. 1/16
4. 1

Correct answer: 1/16

Solution

To find det(BA⁻¹B^T), we can use the properties of determinants. We know that det(ABA^T) = det(A)det(B)det(A^T) = det(A)²det(B) and det(AB⁻¹) = det(A)det(B⁻¹) = det(A)/det(B). Given that both determinants equal 8, we can derive that det(B) = 2det(A). Using these relationships, we can compute det(BA⁻¹B^T) and find that it equals 1/16.

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