`2x - y - 2z=2, x-2y+z=-4, x+y + lamdaz=4` then the value of `lamda` such that system of equations has no solution, is

Options

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

Answer

To find the value of \( \lambda \) such that the system of equations has no solution, we start with the given equations:

1. \( 2x - y - 2z = 2 \)
2. \( x - 2y + z = -4 \)
3. \( x + y + \lambda z = 4 \)

### Step 1: Write the system in matrix form
We can represent the system of equations in the form of a matrix:

\[ \begin{bmatrix} 2 & -1 & -2 \\ 1 & -2 & 1 \\ 1 & 1 & \lambda \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 2 \\ -4 \\ 4 \end{bmatrix} \]

### Step 2: Find the determinant of the coefficient matrix
For the system to have no solution, the determinant of the coefficient matrix must be zero. We calculate the determinant:

\[ D = \begin{vmatrix} 2 & -1 & -2 \\ 1 & -2 & 1 \\ 1 & 1 & \lambda \end{vmatrix} \]

### Step 3: Calculate the determinant using cofactor expansion
We can expand the determinant along the first row:

\[ D = 2 \begin{vmatrix} -2 & 1 \\ 1 & \lambda \end{vmatrix} - (-1) \begin{vmatrix} 1 & 1 \\ 1 & \lambda \end{vmatrix} - 2 \begin{vmatrix} 1 & -2 \\ 1 & 1 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

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

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

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

Substituting these back into the determinant expression:

\[ D = 2(-2\lambda - 1) + (\lambda - 1) - 2(3) \]

### Step 4: Simplify the determinant
Now simplifying:

\[ D = -4\lambda - 2 + \lambda - 1 - 6 \]
\[ D = -4\lambda + \lambda - 2 - 1 - 6 \]
\[ D = -3\lambda - 9 \]

### Step 5: Set the determinant to zero
To find \( \lambda \) such that the system has no solution, we set the determinant equal to zero:

\[ -3\lambda - 9 = 0 \]

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

\[ -3\lambda = 9 \]
\[ \lambda = -3 \]

### Final Answer
The value of \( \lambda \) such that the system of equations has no solution is:

\[ \lambda = -3 \]