If A = [ cc 2 & 1 -1 & 2 ], B = [ ll 1 & 0 1 & 1 ], C = ABA ^ T a… - Vidyadip

MCQ

$\text { If } A =\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], B =\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], C = \text{ABA} ^{ T }$ and $ X$
$= A ^{ T } C ^2 A , $ then operatorname det $X$ is equal to :

A
243
B
729
C
27
D
891

✓

Answer

$A=\left[\begin{array}{cc}\sqrt{2} & 1 \\-1 & \sqrt{2} \end{array}\right] \Rightarrow \operatorname{det}(A)=3$
$B=\left[\begin{array}{ll}1 & 0 \\1 & 1 \end{array}\right] \Rightarrow \operatorname{det}(B)=1$
Now $C = \text{ABA} ^{ T }$
$ \Rightarrow \operatorname{det}( C )=(\operatorname{det}( A ))^2 x \operatorname{det}( B )$
$|C|=9$
$\text { Now }|X|=\left|A^{T} C^2 A\right|$
$=\left|A^{T}\right||C|^2|A|$
$=|A|^2|C|^2$
$=9 \times 81$
$=729$

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