The value of a fro which the system of equations `ax+by+z=0,x+ay+z=0,x+y+z=0`
posses non zero solutions are given by

Options

A. `1,2`
B. `1,-1`
C. `1`
D. None of these

Answer

To find the value of \( a \) for which the system of equations

\[ \begin{align*} 1. & \quad ax + by + z = 0 \\ 2. & \quad x + ay + z = 0 \\ 3. & \quad x + y + z = 0 \end{align*} \]

possesses non-zero solutions, we can represent the system in matrix form and find the determinant of the coefficient matrix. The determinant must be equal to zero for the system to have non-trivial (non-zero) solutions.

### Step 1: Write the coefficient matrix

The coefficient matrix \( A \) for the system of equations is:

\[ A = \begin{pmatrix} a & b & 1 \\ 1 & a & 1 \\ 1 & 1 & 1 \end{pmatrix} \]

### Step 2: Calculate the determinant of the matrix

To find the determinant of matrix \( A \), we can use the formula for the determinant of a 3x3 matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix, we have:

\[ \text{det}(A) = a \begin{vmatrix} a & 1 \\ 1 & 1 \end{vmatrix} - b \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} + 1 \begin{vmatrix} 1 & a \\ 1 & 1 \end{vmatrix} \]

Calculating the minors:

1. \( \begin{vmatrix} a & 1 \\ 1 & 1 \end{vmatrix} = a \cdot 1 - 1 \cdot 1 = a - 1 \)
2. \( \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} = 1 \cdot 1 - 1 \cdot 1 = 0 \)
3. \( \begin{vmatrix} 1 & a \\ 1 & 1 \end{vmatrix} = 1 \cdot 1 - a \cdot 1 = 1 - a \)

Substituting these values back into the determinant formula:

\[ \text{det}(A) = a(a - 1) - b(0) + 1(1 - a) \]

This simplifies to:

\[ \text{det}(A) = a(a - 1) + (1 - a) \]

### Step 3: Set the determinant to zero

For the system to have non-zero solutions, we need:

\[ a(a - 1) + (1 - a) = 0 \]

Expanding this gives:

\[ a^2 - a + 1 - a = 0 \implies a^2 - 2a + 1 = 0 \]

### Step 4: Factor the quadratic equation

This can be factored as:

\[ (a - 1)^2 = 0 \]

### Step 5: Solve for \( a \)

Setting the factor equal to zero gives:

\[ a - 1 = 0 \implies a = 1 \]

### Conclusion

The value of \( a \) for which the system of equations possesses non-zero solutions is:

\[ \boxed{1} \]