If `S` is the set of distinct values of `' b` for which the following system of linear equations `x+y+z=1` `x+a y+z=1` `a x+b y+z=0` has no solution, then `S` is : a finite set containing two or more elements (2) a singleton an empty set (4) an infinite set

Options

A. a singleton set
B. an empty set
C. an infinite set
D. a finite set containing two or more elements

Explanation

To determine the set \( S \) of distinct values of \( b \) for which the given system of linear equations has no solution, we can analyze the equations step by step.

### Step-by-Step Solution:

1. **Write down the system of equations**:
\[ \begin{align*} (1) & \quad x + y + z = 1 \\ (2) & \quad x + ay + z = 1 \\ (3) & \quad ax + by + z = 0 \end{align*} \]

2. **Formulate the augmented matrix**:
The augmented matrix for the system is:
\[ \begin{bmatrix} 1 & 1 & 1 & | & 1 \\ 1 & a & 1 & | & 1 \\ a & b & 1 & | & 0 \end{bmatrix} \]

3. **Calculate the determinant of the coefficient matrix**:
The coefficient matrix is:
\[ \begin{bmatrix} 1 & 1 & 1 \\ 1 & a & 1 \\ a & b & 1 \end{bmatrix} \]
We will calculate its determinant using the formula for a \( 3 \times 3 \) matrix:
\[ \text{det} = 1 \cdot (a \cdot 1 - 1 \cdot b) - 1 \cdot (1 \cdot 1 - 1 \cdot a) + 1 \cdot (1 \cdot b - a \cdot 1) \]
Simplifying this gives:
\[ \text{det} = a - b - (1 - a) + (b - a) = 2a - 1 - 2a + b = b - 1 \]

4. **Set the determinant equal to zero for no solution**:
For the system to have no solution, the determinant must be zero:
\[ b - 1 = 0 \implies b = 1 \]

5. **Determine the conditions for \( a \)**:
When \( b = 1 \), we substitute \( b \) back into the equations. The equations become:
\[ \begin{align*} x + y + z &= 1 \\ x + ay + z &= 1 \\ ax + y + z &= 0 \end{align*} \]
If \( a = 1 \), the first two equations are identical, leading to a contradiction with the third equation.

6. **Conclusion about the set \( S \)**:
The only value of \( b \) that leads to no solution is \( b = 1 \). Therefore, the set \( S \) is a singleton set containing the single value \( \{1\} \).

### Final Answer:
The set \( S \) is a singleton.