Question

Let $A$ be a $3 \times 3$ matrix and $\det(A) = 2$. If $n = \det(\text{adj}(\text{adj}(\ldots(\text{adj}(A))\ldots)))$ with adjoint applied 2024 times, then the remainder when $n$ is divided by 9 is equal to \\_\\_\\_\\_\\_.

Answer 1

$2^{2024} = (2^2)^{2022} = 4 \cdot (8)^{674} = 4 \cdot (9 - 1)^{674}.$

Applying modulo 9, we get:

$2^{2024} \equiv 4 \pmod{9}.$

Thus,

$2^{2024} = 9m + 4, \quad m \text{ is even}.$

Now, consider $2^{9m+4}$:

$2^{9m+4} = 16 \cdot (2^3)^{3m} \equiv 16 \pmod{9}.$

Thus,

$= 7.$

Therefore, the answer is:

7\.