Question

If \( A=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], B=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], C=A B A^{T} \) and \( X=A^{\top} C^{2} A \), then \( \operatorname{det} X \) is equal to : (1) 243 (2) 891 (3) 729

Answer 1

answer is 729 as follows

Answer 2

Given C = ABA^T   and X = A^TC²A 

AA^T = 3 I -- Eqn 1

X =  A^TC²A  =  A^TC C A =  A^T (ABA^T )( ABA^T ) A = A^TA ( B ) A^TA ( B ) A^TA  = 3I (B) 3I (B) 3I

X = 27(B) (B)

det B = 1 -- Eqn 2

using det  properties

det X = 27² (det B )( det B) = 729 (1 ) (1 ) = 729