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If A is a 3 × 3 matrix and \left | A \right |=2 , then   3 \operatorname{adj}\left(|3 \mathrm{~A}| \mathrm{A}^2\right) \mid is equal to :

Option: 1

3^{12}\: .\: 6^{10}


Option: 2

3^{11}\: .\: 6^{10}


Option: 3

3^{12}\: .\: 6^{11}


Option: 4

3^{10}\: .\: 6^{11}


Answers (1)

best_answer

Given  |\mathrm{A}|=2

Now \left|3 \operatorname{adj}\left(|3 \mathrm{~A}| \mathrm{A}^2\right)\right|

\begin{aligned} & |3 \mathrm{~A}|=3^3 \cdot|\mathrm{A}| \\ & =3^3 \cdot(2) \\ & \text { Adj. }\left(|3 \mathrm{~A}| \mathrm{A}^2\right)=\operatorname{adj}\left\{\left(3^3 \cdot 2\right) \mathrm{A}^2\right\} \\ & =\left(2 \cdot 3^3\right)^2(\operatorname{adj~A})^2 \\ & =2^2 \cdot 3^6 \cdot(\operatorname{adj~A})^2 \end{aligned}

\begin{aligned} \left|3 \operatorname{adj}\left(|3 \mathrm{~A}| \mathrm{A}^2\right)\right| & =\left|2^2 \cdot 3 \cdot 3^6(\operatorname{adj} \mathrm{A})^2\right| \\ & =\left(2^2 \cdot 3^7\right)^3 \mid \text { adj }\left.\mathrm{A}\right|^2 \\ & =2^6 \cdot 3^{21}\left(|\mathrm{~A}|^2\right)^2 \\ & =2^6 \cdot 3^{21}\left(2^2\right)^2 \\ & =2^{10} \cdot 3^{21} \\ & =2^{10} \cdot 3^{10} \cdot 3^{11} \\ \left|3 \operatorname{adj}\left(|3 \mathrm{~A}| \mathrm{A}^2\right)\right| & =6^{10} \cdot 3^{11} \end{aligned}

 

Posted by

Sanket Gandhi

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