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

MCQ

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

A
$243$
✓
$729$
C
$27$
D
$891$

✓

Answer

Correct option: B.

$729$

b
$\begin{aligned} & A=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right] \Rightarrow \operatorname{det}(A)=3 \\ & B=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right] \Rightarrow \operatorname{det}(B)=1\end{aligned}$

Now $C=A B A^T \Rightarrow \operatorname{det}(C)=(\operatorname{dct}(A))^2 x \operatorname{det}(B)$

$|C|=9$

$\text { Now }|X|=\left|A^T C^2 A\right|$

$=\left|A^T\right||C|^2|A|$

$=|A|^2|C|^2$

$=9 \times 81$

$=729$

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