Consider the system of equations :
x + y + z = 0
`alpha x + beta y + gamma z = `0
`alpha^(2) x + beta^(2) y + gamma^(2) z = 0 `
then the system of equations has

Options

A. a unique solution for all values of `alpha, beta , gamma `
B. infinite number of solutions if any two of `alpha, beta , gamma` are equal
C. a unique solution if `alpha, beta , gamma ` are distinct
D. more than one,but finite number of solutions depending on values of `alpha, beta, gamma `

Answer

To solve the given system of equations:

1. **Write down the equations:**
\[ \begin{align*} 1. & \quad x + y + z = 0 \\ 2. & \quad \alpha x + \beta y + \gamma z = 0 \\ 3. & \quad \alpha^2 x + \beta^2 y + \gamma^2 z = 0 \end{align*} \]

2. **Form the coefficient matrix and set up the determinant:**
The coefficient matrix \( A \) can be represented as:
\[ A = \begin{bmatrix} 1 & 1 & 1 \\ \alpha & \beta & \gamma \\ \alpha^2 & \beta^2 & \gamma^2 \end{bmatrix} \]
The determinant of this matrix, denoted as \( \Delta \), is:
\[ \Delta = \begin{vmatrix} 1 & 1 & 1 \\ \alpha & \beta & \gamma \\ \alpha^2 & \beta^2 & \gamma^2 \end{vmatrix} \]

3. **Calculate the determinant using expansion:**
We can expand the determinant along the first row:
\[ \Delta = 1 \cdot \begin{vmatrix} \beta & \gamma \\ \beta^2 & \gamma^2 \end{vmatrix} - 1 \cdot \begin{vmatrix} \alpha & \gamma \\ \alpha^2 & \gamma^2 \end{vmatrix} + 1 \cdot \begin{vmatrix} \alpha & \beta \\ \alpha^2 & \beta^2 \end{vmatrix} \]

This results in:
\[ \Delta = (\beta \cdot \gamma^2 - \gamma \cdot \beta^2) - (\alpha \cdot \gamma^2 - \gamma \cdot \alpha^2) + (\alpha \cdot \beta^2 - \beta \cdot \alpha^2) \]

Simplifying this gives:
\[ \Delta = \gamma^2(\beta - \alpha) + \alpha^2(\gamma - \beta) + \beta^2(\alpha - \gamma) \]

4. **Analyze the determinant:**
- If \( \alpha, \beta, \gamma \) are distinct, then \( \Delta \neq 0 \) which implies a unique solution.
- If any two of \( \alpha, \beta, \gamma \) are equal, then \( \Delta = 0 \), which implies infinitely many solutions.
- If all three are equal, \( \Delta = 0 \) also leads to infinitely many solutions.

5. **Conclusion:**
- The system of equations has:
- A unique solution if \( \alpha, \beta, \gamma \) are distinct.
- Infinitely many solutions if any two (or all three) of \( \alpha, \beta, \gamma \) are equal.