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JEE Main 3-April-2025 Paper - Shift 2 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 3-April-2025 Paper - Shift 24 Marks
Question
Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $\mathrm{A}=\left[\begin{array}{ccc}\lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2\end{array}\right],|\mathrm{A}|=-1$. Let B be the inverse of the matrix $\operatorname{adj}\left(\mathrm{A} \operatorname{adj}\left(\mathrm{A}^{2}\right)\right)$. Then $|(\lambda B+1)|$ is equal to __________.
✓

Answer

38
$|A|=\left|\begin{array}{ccc}\lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2\end{array}\right|=-1$
$\lambda(16)-2(-34)+3(-39)=-1$
$16 \lambda=48 \Rightarrow \lambda=3$
$\mathrm{B}^{-1}=\operatorname{adj}\left(\mathrm{A} \cdot \operatorname{adj}\left(\mathrm{A}^{2}\right)\right)$
Let $\mathrm{C}=\mathrm{A} \cdot \operatorname{adj}\left(\mathrm{A}^{2}\right)$
$A C=A^{2} \operatorname{adj}\left(A^{2}\right)=|A|^{2} \cdot I=I \Rightarrow C=A^{-1}$
Now $\mathrm{B}^{-1}=\operatorname{adj}\left(\mathrm{A}^{-1}\right)=\mathrm{B}=\operatorname{adj}(\mathrm{A})$
Now $\lambda \mathrm{B}+\mathrm{I} \Rightarrow 3 \mathrm{~B}+\mathrm{I}$
Let $\mathrm{P}=3 \mathrm{~B}+\mathrm{I}$
P $=3 \operatorname{adj}(\mathrm{~A})+\mathrm{I}$
$\mathrm{AP}=3 \operatorname{Aadj}(\mathrm{~A})+\mathrm{A}$
$\mathrm{AP}=3|\mathrm{~A}| \cdot \mathrm{I}+\mathrm{A}$
$\mathrm{AP}=\mathrm{A}-3 \mathrm{I}$
$|\mathrm{AP}|=|\mathrm{A}-3 \mathrm{I}|$
$|\mathrm{A}| .|\mathrm{P}|=\left|\begin{array}{ccc}0 & 2 & 3 \\ 4 & 2 & 6 \\ 7 & -1 & -1\end{array}\right|=38$
$|\mathrm{P}|=-38$

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