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JEE MAIN 2025
22-01-2025 SHIFT-2
Question
For a $3 \times 3$ matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A|=\frac{1}{2}$ and trace $(A)=3$. If $B=\operatorname{adj}(\operatorname{adj}(2 A))$, then the value of $|B|+$ trace $(B)$ equals:
Select the correct option:
A
280
B
132
C
56
D
174
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
\begin{aligned} & |\mathrm{A}|=\frac{1}{2}, \operatorname{trace}(\mathrm{~A})=3, \mathrm{~B}=\operatorname{adj}(\operatorname{adj}(2 \mathrm{~A}))=|2 \mathrm{~A}|^{\mathrm{n}-2}(2 \mathrm{~A}) \\ & \mathrm{n}=3, \mathrm{~B}=|2 \mathrm{~A}|(2 \mathrm{~A})=2^3 \cdot|\mathrm{~A}|(2 \mathrm{~A})=8 \mathrm{~A} \\ & |\mathrm{~B}|=|8 \mathrm{~A}|=8^3 \cdot|\mathrm{~A}|=2^8=256 \\ & \operatorname{trace}(\mathrm{~B})=8 \operatorname{trace}(\mathrm{~A})=24 \\ & |\mathrm{~B}|+\operatorname{trace}(\mathrm{B})=280 \end{aligned}
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