Matrices Question 9
Question 9 - 2024 (31 Jan Shift 2)
Let $A$ be a $3 \times 3$ matrix and $\operatorname{det}(A)=2$. If
$$ n=\operatorname{det}(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots \ldots(\operatorname{adj} A)} _{2024-\text { times }}))) $$
Then the remainder when $n$ is divided by 9 is equal to
Show Answer
Answer (7)
Solution
$|A|=2$
$\underbrace{\operatorname{adj}(\operatorname{adj}(\operatorname{adj} \ldots \ldots(a)))} _{2024 \text { times }}=|A|^{(n-1)^{2024}}$
$=|A|^{2024}$
$=2^{2024}$
$2^{2024}=\left(2^{2}\right) 2^{2022}=4(8)^{674}=4(9-1)^{674}$
$\Rightarrow 2^{2024} \equiv 4(\bmod 9)$
$\Rightarrow 2^{2024} \equiv 9 m+4, m \leftarrow$ even
$2^{9 m+4} \equiv 16 \cdot\left(2^{3}\right)^{3 m} \equiv 16(\bmod 9)$
$\equiv 7$
