Suppose `a, b, c, in R ` and `abc = 1, if A = [[3a, b, c ],[b, 3c, a ],[c, a, 3b]]` is such that `A ^(T) A = 4 ^(1//3) I and abs(A) gt 0, ` the value of `a^(3) + b^(3) + c^(3)` is

Explanation

To solve the problem step-by-step, we will analyze the given matrix \( A \) and the conditions provided.

### Step 1: Write down the matrix \( A \)
The matrix \( A \) is given as:
\[ A = \begin{bmatrix} 3a & b & c \\ b & 3c & a \\ c & a & 3b \end{bmatrix} \]

### Step 2: Use the property of the transpose
We know that \( A^T A = 4^{1/3} I \). The determinant of \( A^T A \) can be expressed as:
\[ \text{det}(A^T A) = \text{det}(A)^2 = (4^{1/3})^3 = 4 \]
This implies:
\[ \text{det}(A)^2 = 4 \implies \text{det}(A) = 2 \quad (\text{since } |A| > 0) \]

### Step 3: Calculate the determinant of \( A \)
The determinant of a \( 3 \times 3 \) matrix can be calculated using the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For our matrix \( A \):
\[ \text{det}(A) = 3a \begin{vmatrix} 3c & a \\ a & 3b \end{vmatrix} - b \begin{vmatrix} b & a \\ c & 3b \end{vmatrix} + c \begin{vmatrix} b & 3c \\ c & a \end{vmatrix} \]

Calculating the minors:
1. \( \begin{vmatrix} 3c & a \\ a & 3b \end{vmatrix} = 9bc - a^2 \)
2. \( \begin{vmatrix} b & a \\ c & 3b \end{vmatrix} = 3b^2 - ac \)
3. \( \begin{vmatrix} b & 3c \\ c & a \end{vmatrix} = ab - 3c^2 \)

Putting these back into the determinant:
\[ \text{det}(A) = 3a(9bc - a^2) - b(3b^2 - ac) + c(ab - 3c^2) \]
Expanding this gives:
\[ = 27abc - 3a^3 - 3b^3 + 2abc - 3c^3 \]
Combining like terms:
\[ = 29abc - 3(a^3 + b^3 + c^3) \]

### Step 4: Set the determinant equal to 2
From Step 2, we have:
\[ 29abc - 3(a^3 + b^3 + c^3) = 2 \]
Given \( abc = 1 \):
\[ 29(1) - 3(a^3 + b^3 + c^3) = 2 \]
This simplifies to:
\[ 29 - 3(a^3 + b^3 + c^3) = 2 \]

### Step 5: Solve for \( a^3 + b^3 + c^3 \)
Rearranging the equation:
\[ 3(a^3 + b^3 + c^3) = 29 - 2 \]
\[ 3(a^3 + b^3 + c^3) = 27 \]
Dividing by 3:
\[ a^3 + b^3 + c^3 = 9 \]

### Final Answer
The value of \( a^3 + b^3 + c^3 \) is:
\[ \boxed{9} \]