If A = [ 2 1 - 1 2 ] , ? B = [ 1 0 1 1 ] , ? C = A B A T and X = A T C 2 A , then det X is equal to [2024]

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Solution

Q.
If $A = [\begin{matrix} \sqrt{2} & 1 \\ - 1 & \sqrt{2} \end{matrix}], B = [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}], C = A B A^{T}$ and $X = A^{T} C^{2} A,$ then det $X$ is equal to [2024]

1 729

2 243

3 891

4 27

Ans.
(1)

Given, $X = A^{T} C^{2} A$ and $C = A B A^{T}$

$\Rightarrow | X | = | A^{T} C^{2} A | \Rightarrow | X | = | A | | C |^{2} | A^{T} |$

$\Rightarrow | C | = | A B A^{T} | = | A | | B | | A^{T} |$

As $| A | = 3, | A^{T} | = 3, | B | = 1, | C | = 3 \times 1 \times 3 = 9$

$∴$ $| X | = 3 \times 9^{2} \times 3 = 9 \times 81 = 729$

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