JEE 2023 — Mathematics PYQ

Given that $A$ is a matrix of order $3\times 3$ and $|A|=2$
$\Rightarrow |3A|=3^3|A|(\because |kA|=k^n|A|)$
$=3^3\times 2$

Now $\text{adj}(|3A|A^2)=\text{adj}(3^3\times 2A^2)$
$=(3^3\times 2{)}^{3-1}(\text{adj }A{)}^2$
$=3^6\times 2^2(\text{adj }A{)}^2$

and $|3\text{ adj}(|3A|A^2)|=|3\times 3^6\times 2^2(\text{adj }A{)}^2|$
$=(3^7\times 2^2{)}^3|\text{adj }A{|}^2$
$=(3^7\times 2^2{)}^3(|A{|}^2{)}^2$
$=3^{21}\times 2^6\times (2^2{)}^2$
$=3^{11}\times 6^{10}$