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 (integer)
7
Solution
<p>$$\begin{aligned}
& |\mathrm{A}|=2 \\
& \underbrace{\operatorname{adj}(\operatorname{adj}(\operatorname{adj} \ldots . .(\mathrm{a})))}_{2024 \text { times }}=|\mathrm{A}|^{(\mathrm{n}-1)^{2024}} \\
& =|\mathrm{A}|^{2024} \\
& =2^{2^{2024}}
\end{aligned}$$</p>
<p>$$\begin{aligned}
& 2^{2024}=\left(2^2\right) 2^{2022}=4(8)^{674}=4(9-1)^{674} \\
& \Rightarrow 2^{2024} \equiv 4(\bmod 9) \\
& \Rightarrow 2^{2024} \equiv 9 \mathrm{~m}+4, \mathrm{~m} \leftarrow \text { even } \\
& 2^{9 \mathrm{~m}+4} \equiv 16 \cdot\left(2^3\right)^{3 \mathrm{~m}} \equiv 16(\bmod 9) \\
& \quad \equiv 7
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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