JEE MAIN - Mathematics (2024 - 31st January Evening Shift - No. 29)
Let A be a $$3 \times 3$$ matrix and $$\operatorname{det}(A)=2$$. If $$n=\operatorname{det}(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots . .(\operatorname{adj} A))}_{2024-\text { times }}))$$, then the remainder when $$n$$ is divided by 9 is equal to __________.
Answer
7
Explanation
$$\begin{aligned} & |\mathrm{A}|=2 \\ & \underbrace{\operatorname{adj}(\operatorname{adj}(\operatorname{adj} \ldots . .(\mathrm{a})))}_{2024 \text { times }}=|\mathrm{A}|^{(\mathrm{n}-1)^{2024}} \\ & =|\mathrm{A}|^{2024} \\ & =2^{2^{2024}} \end{aligned}$$
$$\begin{aligned} & 2^{2024}=\left(2^2\right) 2^{2022}=4(8)^{674}=4(9-1)^{674} \\ & \Rightarrow 2^{2024} \equiv 4(\bmod 9) \\ & \Rightarrow 2^{2024} \equiv 9 \mathrm{~m}+4, \mathrm{~m} \leftarrow \text { even } \\ & 2^{9 \mathrm{~m}+4} \equiv 16 \cdot\left(2^3\right)^{3 \mathrm{~m}} \equiv 16(\bmod 9) \\ & \quad \equiv 7 \end{aligned}$$
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