Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
Let A and B be two invertible matrices of order 3 x 3. If det(ABA^T) = 8 and det(AB^-1) = 8, then det(BA^-1 B^T) is equal to :-
Step-by-Step Solution
Key Concept: Use properties of determinants: det(AB) = det(A)det(B), det(A^T) = det(A), det(A^-1) = 1/det(A).
Given det(ABA^T) = 8 => det(A)det(B)det(A^T) = 8 => det(A)^2 det(B) = 8. Also det(AB^-1) = 8 => det(A)/det(B) = 8 => det(A) = 8 det(B). Substituting this into the first equation: (8 det(B))^2 det(B) = 8 => 64 det(B)^3 = 8 => det(B)^3 = 1/8 => det(B) = 1/2. Then det(A) = 8(1/2) = 4. We need to find det(BA^-1 B^T) = det(B)det(A^-1)det(B^T) = det(B) * (1/det(A)) * det(B) = det(B)^2 / det(A) = (1/2)^2 / 4 = (1/4) / 4 = 1/16.
Correct Answer: 2