The value of the determinant : `|(1-alpha,alpha-alpha^(2),alpha^(2)),(1-beta,beta-beta^(2),beta^(2)),(1-gamma,gamma-gamma^(2),gamma^(2))|` is equal to :

Options

A. A) `(alpha - beta) (beta - gamma) (alpha - gamma)`
B. B) `(alpha - beta) (beta - gamma) (gamma - alpha)`
C. C) `(alpha - beta) (beta - gamma) (gamma - alpha) (alpha + beta + gamma)`
D. D) 0

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} 1 - \alpha & \alpha - \alpha^2 & \alpha^2 \\ 1 - \beta & \beta - \beta^2 & \beta^2 \\ 1 - \gamma & \gamma - \gamma^2 & \gamma^2 \end{vmatrix} \]

we will perform a series of column and row operations.

### Step 1: Column Operations
We will first perform the column operation \( C_1 \leftarrow C_1 + C_2 + C_3 \) to simplify the first column.

\[ C_1 = (1 - \alpha) + (\alpha - \alpha^2) + \alpha^2 = 1 \]

So, the first column becomes:

\[ C_1 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \]

The determinant now looks like this:

\[ D = \begin{vmatrix} 1 & \alpha - \alpha^2 & \alpha^2 \\ 1 & \beta - \beta^2 & \beta^2 \\ 1 & \gamma - \gamma^2 & \gamma^2 \end{vmatrix} \]

### Step 2: Further Column Operations
Next, we apply the operation \( C_2 \leftarrow C_2 + C_3 \):

\[ C_2 = (\alpha - \alpha^2) + \alpha^2 = \alpha \]
\[ C_2 = (\beta - \beta^2) + \beta^2 = \beta \]
\[ C_2 = (\gamma - \gamma^2) + \gamma^2 = \gamma \]

Now, the determinant becomes:

\[ D = \begin{vmatrix} 1 & \alpha & \alpha^2 \\ 1 & \beta & \beta^2 \\ 1 & \gamma & \gamma^2 \end{vmatrix} \]

### Step 3: Row Operations
Now we will perform row operations. We will subtract the first row from the second and third rows:

\[ R_2 \leftarrow R_2 - R_1 \quad \text{and} \quad R_3 \leftarrow R_3 - R_1 \]

This gives us:

\[ D = \begin{vmatrix} 1 & \alpha & \alpha^2 \\ 0 & \beta - \alpha & \beta^2 - \alpha^2 \\ 0 & \gamma - \alpha & \gamma^2 - \alpha^2 \end{vmatrix} \]

### Step 4: Expanding the Determinant
Now we can expand the determinant along the first column:

\[ D = 1 \cdot \begin{vmatrix} \beta - \alpha & \beta^2 - \alpha^2 \\ \gamma - \alpha & \gamma^2 - \alpha^2 \end{vmatrix} \]

### Step 5: Simplifying the 2x2 Determinant
Now, we compute the 2x2 determinant:

\[ D = (\beta - \alpha)((\gamma^2 - \alpha^2)) - (\gamma - \alpha)(\beta^2 - \alpha^2) \]

Using the identity \( a^2 - b^2 = (a-b)(a+b) \):

\[ D = (\beta - \alpha)(\gamma - \alpha)(\gamma + \alpha) - (\gamma - \alpha)(\beta - \alpha)(\beta + \alpha) \]

### Step 6: Factoring Out Common Terms
We can factor out \( (\beta - \alpha)(\gamma - \alpha) \):

\[ D = (\beta - \alpha)(\gamma - \alpha) \left( (\gamma + \alpha) - (\beta + \alpha) \right) \]
\[ D = (\beta - \alpha)(\gamma - \alpha)(\gamma - \beta) \]

### Final Result
Thus, the value of the determinant is:

\[ D = (\alpha - \beta)(\beta - \gamma)(\gamma - \alpha) \]