Let `A=[a_(ij)]_(3xx3)` be such that `a_(ij)=[{:(3, " when",hati=hatj),(0,,hati ne hatj):}" then " {("det (adj(adj A))")/(5)}` equals :
( where {.} denotes fractional part function )

Options

A. `(2)/(5)`
B. `(1)/(5)`
C. `(2)/(3)`
D. `(1)/(3)`

Answer

To solve the problem, we need to find the fractional part of \(\frac{\text{det}(\text{adj}(\text{adj} A))}{5}\) for the given matrix \(A\). Let's break it down step by step.

### Step 1: Define the Matrix \(A\)
The matrix \(A\) is defined such that:
- \(a_{ij} = 3\) when \(i = j\) (diagonal elements)
- \(a_{ij} = 0\) when \(i \neq j\) (off-diagonal elements)

Thus, the matrix \(A\) can be written as:
\[ A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix} \]

### Step 2: Calculate the Determinant of \(A\)
The determinant of a diagonal matrix is the product of its diagonal entries. Therefore, we have:
\[ \text{det}(A) = 3 \times 3 \times 3 = 27 \]

### Step 3: Use the Formula for Determinant of the Adjoint
The formula for the determinant of the adjoint of a matrix \(A\) is given by:
\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^{n-1} \]
where \(n\) is the order of the matrix. Since \(A\) is a \(3 \times 3\) matrix, \(n = 3\). Thus:
\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^{3-1} = (27)^{2} = 729 \]

### Step 4: Calculate the Determinant of the Adjoint of the Adjoint
Now we need to calculate \(\text{det}(\text{adj}(\text{adj}(A)))\). We can apply the same formula:
\[ \text{det}(\text{adj}(\text{adj}(A))) = (\text{det}(\text{adj}(A)))^{n-1} = (729)^{2} = 531441 \]

### Step 5: Calculate the Fractional Part
Now we need to find the fractional part of \(\frac{\text{det}(\text{adj}(\text{adj}(A)))}{5}\):
\[ \frac{\text{det}(\text{adj}(\text{adj}(A)))}{5} = \frac{531441}{5} = 106288.2 \]
The integer part is \(106288\) and the fractional part is \(0.2\).

### Step 6: Final Answer
The fractional part function is given by:
\[ \{x\} = x - \lfloor x \rfloor \]
Thus, the fractional part of \(\frac{531441}{5}\) is:
\[ \{ \frac{531441}{5} \} = 0.2 \]

### Conclusion
The final answer is:
\[ \text{Fractional part} = \frac{1}{5} \]