For what value of `lambda`, the system of equations `x+y+z=6,x+2y+3z=10,x+2y+lambdaz=12` is inconsistent

Options

A. `lambda=1`
B. `lambda=2`
C. `lambda=-2`
D. `lambda=3`

Answer

To determine the value of \( \lambda \) for which the system of equations is inconsistent, we need to analyze the given equations:

1. \( x + y + z = 6 \) (Equation 1)
2. \( x + 2y + 3z = 10 \) (Equation 2)
3. \( x + 2y + \lambda z = 12 \) (Equation 3)

### Step 1: Write the system in matrix form
We can represent the system of equations in matrix form as \( A \mathbf{x} = \mathbf{b} \), where:

\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \lambda \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 6 \\ 10 \\ 12 \end{bmatrix} \]

### Step 2: Calculate the determinant of matrix \( A \)
To find the value of \( \lambda \) that makes the system inconsistent, we need to find the determinant of matrix \( A \) and set it equal to zero:

\[ \text{det}(A) = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \lambda \end{vmatrix} \]

### Step 3: Expand the determinant
Using the first row to expand the determinant:

\[ \text{det}(A) = 1 \cdot \begin{vmatrix} 2 & 3 \\ 2 & \lambda \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 3 \\ 1 & \lambda \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 2 \\ 1 & 2 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. \( \begin{vmatrix} 2 & 3 \\ 2 & \lambda \end{vmatrix} = 2\lambda - 6 \)

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

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

Putting it all together:

\[ \text{det}(A) = (2\lambda - 6) - (\lambda - 3) + 0 \]

### Step 4: Simplify the determinant
Now simplify the expression:

\[ \text{det}(A) = 2\lambda - 6 - \lambda + 3 = \lambda - 3 \]

### Step 5: Set the determinant to zero
For the system to be inconsistent, the determinant must equal zero:

\[ \lambda - 3 = 0 \]

### Step 6: Solve for \( \lambda \)
Solving for \( \lambda \):

\[ \lambda = 3 \]

### Conclusion
The value of \( \lambda \) for which the system of equations is inconsistent is \( \lambda = 3 \).

---