What is the value of `|{:(a,b,c),(b,c,a),(c,a,b):}| "if a"^(3) + b^(3) + c^(3) = 0` ?

Options

A. 0
B. 1
C. 3 abc
D. `-3 abc`

Explanation

To find the value of the determinant \( | \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} | \) given that \( a^3 + b^3 + c^3 = 0 \), we can follow these steps:

### Step 1: Write down the determinant
We need to calculate the determinant of the matrix:
\[ D = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} \]

### Step 2: Use the determinant formula
The determinant of a 3x3 matrix can be calculated using the formula:
\[ D = a \begin{vmatrix} c & a \\ a & b \end{vmatrix} - b \begin{vmatrix} b & a \\ c & b \end{vmatrix} + c \begin{vmatrix} b & c \\ c & a \end{vmatrix} \]

### Step 3: Calculate the minors
Now we calculate each of the 2x2 determinants:
1. \( \begin{vmatrix} c & a \\ a & b \end{vmatrix} = cb - a^2 \)
2. \( \begin{vmatrix} b & a \\ c & b \end{vmatrix} = bb - ac = b^2 - ac \)
3. \( \begin{vmatrix} b & c \\ c & a \end{vmatrix} = ba - c^2 \)

### Step 4: Substitute back into the determinant formula
Substituting these values back into the determinant formula gives:
\[ D = a(cb - a^2) - b(b^2 - ac) + c(ba - c^2) \]

### Step 5: Simplify the expression
Expanding this expression:
\[ D = acb - a^3 - b^3 + abc + abc - c^3 \]
Combining like terms:
\[ D = 3abc - (a^3 + b^3 + c^3) \]

### Step 6: Use the given condition
Since we know that \( a^3 + b^3 + c^3 = 0 \), we can substitute this into our expression for \( D \):
\[ D = 3abc - 0 = 3abc \]

### Conclusion
Thus, the value of the determinant is:
\[ \boxed{3abc} \]