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 ⁻¹ ~B ^ T ) is equal to

Step-by-step solution

Let |A|=a,|B|=b . A^ =a |A⁻¹ |= 1 a , |B^ |=b, |B⁻¹ |= 1 b A B A^ =8 |A||B| A^ -8 (1) a b a=8 a² b=8 A B⁻¹|=8 H| |B⁻¹ |=8 a, 1 b =8 From (1) & (2) a=4, b= 1 2 Then, |B A⁻¹ B^ |=|B| |A⁻¹ | |B^ |=b 1 a b= b² a = 1 16