If `A=[a_("ij")]_(4xx4)`, such that `a_("ij")={(2",","when "i=j),(0",","when "i ne j):}`, then `{("det (adj (adj A))")/(7)}` is (where `{*}` represents fractional part function)

Options

A. `1//7`
B. `2//7`
C. `3//7`
D. none of these

Explanation

To solve the problem, we need to follow these steps:

### Step 1: Construct the Matrix A
Given that \( A = [a_{ij}]_{4 \times 4} \) where:
- \( a_{ij} = 2 \) when \( i = j \) (diagonal elements)
- \( a_{ij} = 0 \) when \( i \neq j \) (off-diagonal elements)

Thus, the matrix \( A \) can be represented as:
\[ A = \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{bmatrix} \]

### Step 2: Calculate the Determinant of A
Since \( A \) is a diagonal matrix, the determinant is the product of the diagonal elements:
\[ \text{det}(A) = 2 \times 2 \times 2 \times 2 = 2^4 = 16 \]

### Step 3: Calculate the Determinant of the Adjoint of A
The adjoint of a matrix \( A \) is given by:
\[ \text{adj}(A) = \text{det}(A) \cdot A^{-1} \]
For a \( n \times n \) matrix, the determinant of the adjoint is given by:
\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^{n-1} \]
Here, \( n = 4 \):
\[ \text{det}(\text{adj}(A)) = (16)^{4-1} = 16^3 = 4096 \]

### Step 4: Calculate the Determinant of the Adjoint of the Adjoint of A
Using the same property:
\[ \text{det}(\text{adj}(\text{adj}(A))) = (\text{det}(\text{adj}(A)))^{n-1} \]
Thus,
\[ \text{det}(\text{adj}(\text{adj}(A))) = (4096)^{4-1} = 4096^3 \]
Calculating \( 4096 \):
\[ 4096 = 2^{12} \quad \text{(since } 16 = 2^4 \text{, so } 16^3 = (2^4)^3 = 2^{12}) \]
Thus,
\[ \text{det}(\text{adj}(\text{adj}(A))) = (2^{12})^3 = 2^{36} \]

### Step 5: Calculate the Fractional Part of \(\frac{\text{det}(\text{adj}(\text{adj}(A)))}{7}\)
Now we need to find:
\[ \frac{2^{36}}{7} \]
Calculating \( 2^{36} \):
\[ 2^{36} = 68719476736 \]
Now dividing by 7:
\[ 68719476736 \div 7 = 9817066676 \quad \text{(integer part)} \]
Calculating the remainder:
\[ 68719476736 \mod 7 = 4 \quad \text{(since } 68719476736 = 7 \times 9817066676 + 4\text{)} \]
Thus, we have:
\[ \frac{2^{36}}{7} = 9817066676 + \frac{4}{7} \]

### Step 6: Find the Fractional Part
The fractional part of \(\frac{2^{36}}{7}\) is:
\[ \{ \frac{2^{36}}{7} \} = \frac{4}{7} \]

### Final Answer
Thus, the answer is:
\[ \frac{4}{7} \]