If system of equation ax + y + z = a, x + by + z = b and x + y + cz = c is inconsistent, then which of the following is correct?

Options

A. abc - a - b - c + 2 = 0
B. abc -a - b - c + 3 = 0,a = 1
C. abc - a - b - c + 3 = 0
D. `abc - a - b - c + 2 = 0,a ne 1,b ne 1,c ne 1`

Answer

To determine the conditions under which the given system of equations is inconsistent, we need to analyze the system of equations:

1. \( ax + y + z = a \)
2. \( x + by + z = b \)
3. \( x + y + cz = c \)

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

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

### Step 2: Find the determinant of the coefficient matrix
The determinant of the coefficient matrix must be calculated to check for consistency. The determinant is given by:

\[ D = \begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix} \]

Calculating the determinant using the formula for a 3x3 matrix:

\[ D = a \begin{vmatrix} b & 1 \\ 1 & c \end{vmatrix} - 1 \begin{vmatrix} 1 & 1 \\ 1 & c \end{vmatrix} + 1 \begin{vmatrix} 1 & b \\ 1 & 1 \end{vmatrix} \]

Calculating the 2x2 determinants:

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

2. \( \begin{vmatrix} 1 & 1 \\ 1 & c \end{vmatrix} = c - 1 \)

3. \( \begin{vmatrix} 1 & b \\ 1 & 1 \end{vmatrix} = 1 - b \)

Substituting these back into the determinant:

\[ D = a(bc - 1) - (c - 1) + (1 - b) \]

This simplifies to:

\[ D = abc - a - c + 1 + 1 - b \]
\[ D = abc - a - b - c + 2 \]

### Step 3: Set the determinant equal to zero for inconsistency
For the system to be inconsistent, the determinant must equal zero:

\[ abc - a - b - c + 2 = 0 \]

### Step 4: Analyze the conditions for inconsistency
From the equation \( abc - a - b - c + 2 = 0 \), we can rearrange it to find:

\[ abc = a + b + c - 2 \]

### Step 5: Determine the conditions on \( a, b, c \)
For the system to be inconsistent, we also need to ensure that at least one of the determinants \( \Delta_1 \) or \( \Delta_2 \) (derived from the modified matrices) is non-zero.

If \( a = 1, b = 1, c = 1 \), then the determinant \( D \) becomes zero, leading to a consistent system. Therefore, for inconsistency, we require:

- \( a \neq 1 \)
- \( b \neq 1 \)
- \( c \neq 1 \)

### Conclusion
Thus, the correct answer is that the system of equations is inconsistent if \( a, b, c \) are not equal to 1.