If system of equation
`x+ay+bcz=0`
`x+by+caz=0`
`x+cy+abz=0`
have non zero solution then which of the following is incorrect

Options

A. `ane b ne c ne a`
B. `a ne b = c`
C. `a = b ne c`
D. `a = b = c`

Explanation

To solve the problem, we need to analyze the given system of equations and determine the conditions under which it has non-zero solutions. The equations are:

1. \( x + ay + bcz = 0 \)
2. \( x + by + caz = 0 \)
3. \( x + cy + abz = 0 \)

### Step 1: Write the system in matrix form
We can express the system of equations in matrix form as follows:

\[ \begin{bmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]

### Step 2: Calculate the determinant of the coefficient matrix
For the system to have non-zero solutions, the determinant of the coefficient matrix must be equal to zero. Let's denote the determinant as \( \Delta \):

\[ \Delta = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix} \]

### Step 3: Expand the determinant
Using the determinant formula for a 3x3 matrix, we can expand \( \Delta \):

\[ \Delta = 1 \cdot (b \cdot ab - ca \cdot c) - a \cdot (1 \cdot ab - ca \cdot 1) + bc \cdot (1 \cdot c - b \cdot 1) \]

Calculating this gives:

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

### Step 4: Simplify the determinant
After simplifying, we find:

\[ \Delta = a^2c - b^2c - a^2b + b^2c + c^2b - c^2a \]

This can be factored as:

\[ \Delta = (a - b)(b - c)(c - a) \]

### Step 5: Set the determinant to zero
For the system to have non-zero solutions, we set \( \Delta = 0 \):

\[ (a - b)(b - c)(c - a) = 0 \]

### Step 6: Analyze the conditions
The product is zero if at least one of the factors is zero, which leads to the following conditions:

1. \( a = b \)
2. \( b = c \)
3. \( c = a \)

### Step 7: Identify the incorrect option
The problem asks for the incorrect statement regarding the conditions for non-zero solutions. The only case that does not satisfy the determinant being zero is when \( a \neq b \neq c \neq a \).

Thus, the incorrect option is:

**Option: \( a \neq b \neq c \neq a \)**