ધારોકે $A$ એ કક્ષા $3 × 3$ વાળો શ્રેણિક છે અને $\operatorname{det}(A)=2$ છ. તો $\operatorname{det}\left(\operatorname{det}(A) \operatorname{adj}\left(5 \operatorname{adj}\left(A^{3}\right)\right)\right)=$................
JEE MAIN 2022, Difficult
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$|(\operatorname{det}( A )) \operatorname{adj}(5 \operatorname{adj}( A ))|$
$=\left|2\, \operatorname{adj}\left(5 \operatorname{adj}\left( A ^{3}\right)\right)\right|$
$=2^{3} \operatorname{ladj}\left(5 \operatorname{adj}\left( A ^{3}\right) \|\right.$
$=2^{3} \cdot\left|5 adj \left( A ^{3}\right)\right|^{2}$
$=2^{3}\left(5^{3} \cdot \operatorname{ladj}\left( A ^{3}\right) 1\right)^{2}$
$=2^{3} \cdot 5^{6} \cdot\left|\operatorname{adj} A^{3}\right|^{2}$
$=2^{3} \cdot 5^{6}\left(\left(| Al |^{3}\right)^{2}\right)^{2}$
$=2^{3} \cdot 5^{6} \cdot 2^{12}=2^{15} \times 5^{6}$
$=2^{9} \times 10^{6}$
$=512 \times 10^{6}$.
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