Let A be a matrix of order 3 × 3 and det A = 2 . Then det det ( A ) adj 5 adj A 3 is equal to _____.

Let A be a matrix of order 3×3 and detA=2. Then detdet(A)adj5adjA3 is equal to _____.
  1. 256×106
  2. 1024×106
  3. 512×106
  4. 256×1011

Solution

$|\operatorname{det}(A) \operatorname{adj}(5 \operatorname{adj}(A))|$ $=\left|2 \operatorname{adj}\left(5 \operatorname{adj}\left(A^{3}\right)\right)\right|$ $=2^{3} | \operatorname{adj}\left(5 \operatorname{adj}\left(A^{3}\right) \right)|$ $=2^{3} \cdot\left|5 \operatorname{adj}\left(A^{3}\right)\right|^{2}$ $=2^{3}\left(5^{3} \cdot\left|\operatorname{adj}\left(A^{3}\right)\right|\right)^{2}$ $=2^{3} \cdot 5^{6} \cdot\left|\operatorname{adj} A^{3}\right|^{2}$ $=2^{3} \cdot 5^{6}\left(\left(|A|^{3}\right)^{2}\right)^{2}$ $=2^{3} \cdot 5^{6} \cdot 2^{12}=2^{15} \times 5^{6}$ $=2^{9} \times 10^{6}$ $=512 \times 10^{6}$

Asked in: JEE Main 2022 (28 Jun Shift 1)

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