If the system of equations
`bx + ay = c, cx + az = b, cy + bz = a`
has a unique solution, then

Options

A. `abc = 1`
B. `abc = -2`
C. `abc ne 0`
D. none of these

Answer

To determine the condition for the system of equations \( bx + ay = c \), \( cx + az = b \), and \( cy + bz = a \) to have a unique solution, we need to analyze the determinant of the coefficient matrix formed by the coefficients of \( x \), \( y \), and \( z \).

### Step-by-Step Solution:

1. **Write the System of Equations**:
The given system of equations is:
\[ bx + ay = c \quad (1) \]
\[ cx + az = b \quad (2) \]
\[ cy + bz = a \quad (3) \]

2. **Form the Coefficient Matrix**:
The coefficient matrix \( A \) for the system can be represented as:
\[ A = \begin{bmatrix} b & a & 0 \\ c & 0 & a \\ 0 & b & c \end{bmatrix} \]

3. **Calculate the Determinant**:
To find the condition for a unique solution, we need to calculate the determinant of matrix \( A \) and set it not equal to zero:
\[ \text{det}(A) = \begin{vmatrix} b & a & 0 \\ c & 0 & a \\ 0 & b & c \end{vmatrix} \]

We can calculate this determinant using the cofactor expansion along the first row:
\[ \text{det}(A) = b \begin{vmatrix} 0 & a \\ b & c \end{vmatrix} - a \begin{vmatrix} c & a \\ 0 & c \end{vmatrix} + 0 \]

4. **Calculate the 2x2 Determinants**:
Now we calculate the 2x2 determinants:
- For the first determinant:
\[ \begin{vmatrix} 0 & a \\ b & c \end{vmatrix} = (0 \cdot c) - (a \cdot b) = -ab \]
- For the second determinant:
\[ \begin{vmatrix} c & a \\ 0 & c \end{vmatrix} = (c \cdot c) - (a \cdot 0) = c^2 \]

5. **Substituting Back**:
Now substituting back into the determinant expression:
\[ \text{det}(A) = b(-ab) - a(c^2) = -ab^2 - ac^2 \]

6. **Setting the Determinant Not Equal to Zero**:
For the system to have a unique solution, we require:
\[ -ab^2 - ac^2 \neq 0 \]
This can be simplified to:
\[ ab^2 + ac^2 \neq 0 \]

7. **Final Condition**:
Since \( ab^2 + ac^2 \neq 0 \), we conclude that:
\[ abc \neq 0 \]
This means that \( a \), \( b \), and \( c \) must all be non-zero for the system to have a unique solution.