If A is a matrix of order `3 times 3` and `det A = 2` and `n = det ubrace((adj(adj...........adjA)))_("2024 terms")`, the remainder when n is divided by 9, is

Options

A. 2
B. 4
C. 6
D. 7

Answer

To solve the problem, we need to find the value of \( n = \det(\text{adj}(\text{adj}(\ldots \text{adj}(A) \ldots))) \) where the adjoint is taken 2024 times and \( A \) is a \( 3 \times 3 \) matrix with \( \det(A) = 2 \).

### Step-by-Step Solution:

1. **Understanding the Determinant of the Adjoint:**
The determinant of the adjoint of a matrix \( A \) can be expressed as:
\[ \det(\text{adj}(A)) = \det(A)^{n-1} \]
where \( n \) is the order of the matrix. For a \( 3 \times 3 \) matrix, \( n = 3 \).

2. **Calculating the First Adjoint:**
From the property mentioned above:
\[ \det(\text{adj}(A)) = \det(A)^{3-1} = \det(A)^2 \]
Given \( \det(A) = 2 \):
\[ \det(\text{adj}(A)) = 2^2 = 4 \]

3. **Calculating the Second Adjoint:**
Now, we need to find \( \det(\text{adj}(\text{adj}(A))) \):
\[ \det(\text{adj}(\text{adj}(A))) = \det(\text{adj}(A))^{3-1} = \det(\text{adj}(A))^2 \]
Substituting the value we found:
\[ \det(\text{adj}(\text{adj}(A))) = 4^2 = 16 \]

4. **Calculating the Third Adjoint:**
Next, we calculate \( \det(\text{adj}(\text{adj}(\text{adj}(A)))) \):
\[ \det(\text{adj}(\text{adj}(\text{adj}(A)))) = \det(\text{adj}(\text{adj}(A)))^{3-1} = \det(\text{adj}(\text{adj}(A)))^2 \]
Substituting the value we found:
\[ \det(\text{adj}(\text{adj}(\text{adj}(A)))) = 16^2 = 256 \]

5. **Generalizing for 2024 Adjoint Operations:**
We can see a pattern here. Each time we take the adjoint, we square the previous determinant:
\[ \det(\text{adj}^k(A)) = (2^{2^{k-1}})^{2^{k-1}} = 2^{2^k} \]
For \( k = 2024 \):
\[ n = \det(\text{adj}^{2024}(A)) = 2^{2^{2024}} \]

6. **Finding the Remainder when Divided by 9:**
We need to find \( 2^{2^{2024}} \mod 9 \). To do this, we can use Euler's theorem. Since \( \phi(9) = 6 \):
\[ 2^6 \equiv 1 \mod 9 \]
We need to find \( 2^{2024} \mod 6 \):
\[ 2024 \mod 6 = 4 \]
Thus:
\[ 2^{2024} \equiv 2^4 \mod 9 \]
Calculating \( 2^4 \):
\[ 2^4 = 16 \]
Now, find \( 16 \mod 9 \):
\[ 16 \mod 9 = 7 \]

### Final Answer:
The remainder when \( n \) is divided by 9 is \( \boxed{7} \).