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Text Solution

1`(1)/(4)``(1)/(16)`16

Solution :

Given, `|ABA^(T)| = 8` <br> `rArr |A||B||A^(T)|= 8 " " [because|XY|=|X||Y||` <br> `therefore |A|^(2)|B| = 8" "..(i) [because|A^(T)|=|A|]` <br> `"Also, we have "|AB^(-1)| = 8 rArr |A||B^(-1)| = 8` <br> `rArr " " (|A|)/(|B|) = 8 " "...(ii) [because |A^(-1)| = |A|^(-1) = (1)/(|A|)]` <br> On multiplying Eqs. (i) and (ii), we get <br> `|A|^(3) = 8*8 = 4^(3)` <br> `rArr |A| = 4` <br> `rArr |B| = (|A|)/(8) = (4)/(8) = (1)/(2)` <br> Now, `|BA^(-1)B^(T)| = |B|(1)/(|A|)|B| = ((1)/(2)) (1)/(4) ((1)/(2)) = (1)/(16)`