Let A be a matrix of order 3 3 such that (A) > 0 . If ( 1 2 adj (3 adj (2A)) ) = 2¹³ 3^6 , then the value of (2 adj (2A⁻¹)) is equal to

Step-by-step solution

Let (A) = d . We are given ( 1 2 adj (3 adj (2A)) ) = 2¹³ 3^6 . Using the properties of determinants for a 3 3 matrix X : (kX) = k^3 (X) and ( adj (X)) = ( (X))^2 . Evaluating the inner expressions step-by-step: (2A) = 2^3 (A) = 8d ( adj (2A)) = ( (2A))^2 = (8d)^2 = 64d^2 (3 adj (2A)) = 3^3 ( adj (2A)) = 27 64d^2 = 3^3 2^6 d^2 ( adj (3 adj (2A))) = ( (3 adj (2A)))^2 = (3^3 2^6 d^2)^2 = 3^6 2¹² d^4 Now, applying the scalar multiple property for the outermost factor of 1 2 : ( 1 2 adj (3 adj (2A)) ) = ( 1 2 )^3 3^6 2