JEE Main 2025 — 3 April, Evening Shift — Question 42

$B=\left[\operatorname{adj}\left(A \operatorname{adj}\left(A^{2}\right)\right)\right]^{-1}$

$\operatorname{Adj}\left(A^{2}\right)=(\operatorname{adj} A)^{2} \Rightarrow A \operatorname{adj}\left(A^{2}\right)=A \operatorname{adj}(A) \cdot(\operatorname{adj} A)$

$=A\left(|A| A^{-1}\right)^{2}=|A|^{2}\left(A^{-1}\right)=A^{-1}$

$\Rightarrow \quad B=\left(\operatorname{adj}\left(A^{-1}\right)\right)^{-1}=\left(\left|\left(A^{-1}\right)\right| A\right)^{-1}=\frac{A^{-1}}{-1}=-A^{-1}$

$\Rightarrow B=-A^{-1}$

$|A|=-1=\left|\begin{array}{ccc}\lambda & 2 & 3\\ 4 & 5 & 6\\ 7 & -1 & 2\end{array}\right|=-1 \Rightarrow \lambda=3$

$|3 B+I|=\left|I-3 A^{-1}\right|=\frac{|A|\left|I-3 A^{-1}\right|}{|A|}=\frac{|A-3| \mid}{|A|}$

$=\frac{|A-3 /|}{-1}=\frac{\left|\begin{array}{ccc}0 & 2 & 3\\ 4 & 2 & 6\\ 7 & -1 & -1\end{array}\right|}{-1}=38$

$\Rightarrow|3 B+I|=38$