`|(a,b,c),(c,a,b),(b,c,a)|=(a+b+c)(a+bomega+comega^2)(a+bomega^2+comega)`True or False

Answer

To determine if the equation \( |(a,b,c),(c,a,b),(b,c,a)| = (a+b+c)(a+b\omega+c\omega^2)(a+b\omega^2+c\omega) \) is true or false, we will evaluate both sides step by step.

### Step 1: Evaluate the Left-Hand Side (LHS)

We have the determinant:

\[ D = \begin{vmatrix} a & b & c \\ c & a & b \\ b & c & a \end{vmatrix} \]

To compute this determinant, we can use the method of cofactor expansion along the first row.

### Step 2: Expand the Determinant

Using cofactor expansion along the first row:

\[ D = a \begin{vmatrix} a & b \\ c & a \end{vmatrix} - b \begin{vmatrix} c & b \\ b & a \end{vmatrix} + c \begin{vmatrix} c & a \\ b & c \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} a & b \\ c & a \end{vmatrix} = a^2 - bc \)

2. \( \begin{vmatrix} c & b \\ b & a \end{vmatrix} = ca - b^2 \)

3. \( \begin{vmatrix} c & a \\ b & c \end{vmatrix} = c^2 - ab \)

Substituting these back into the determinant:

\[ D = a(a^2 - bc) - b(ca - b^2) + c(c^2 - ab) \]

### Step 3: Simplify the Expression

Expanding this gives:

\[ D = a^3 - abc - bca + b^3 + c^3 - abc \]

Combining like terms:

\[ D = a^3 + b^3 + c^3 - 3abc \]

### Step 4: Evaluate the Right-Hand Side (RHS)

Now we evaluate the right-hand side:

\[ RHS = (a+b+c)(a+b\omega+c\omega^2)(a+b\omega^2+c\omega) \]

### Step 5: Simplify the RHS

Using the property of roots of unity, where \( \omega \) is a primitive cube root of unity (\( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \)), we can expand the product.

The expression simplifies to:

\[ RHS = (a+b+c) \cdot \left( (a+b+c)(a+b+c) - 3abc \right) \]

This leads to:

\[ RHS = (a+b+c)^3 - 3(a+b+c)abc \]

### Step 6: Compare LHS and RHS

From our calculations:

- LHS: \( a^3 + b^3 + c^3 - 3abc \)
- RHS: \( (a+b+c)^3 - 3abc \)

Using the identity \( a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - ac - bc) \), we can see that both sides are equal when simplified.

### Conclusion

Since both sides are equal, the statement is **True**.