Given , 2x-y+2z=2, x-2y+z=-4, x+y+`lambda`z=4, then the value of `lambda` such that the given system of equations has no solution is :

Options

A. 3
B. 1
C. 0
D. `-3`

Answer

To solve the given system of equations for the value of \( \lambda \) such that there is no solution, we can follow these steps:

### Step 1: Write down the equations
The given equations are:
1. \( 2x - y + 2z = 2 \)
2. \( x - 2y + z = -4 \)
3. \( x + y + \lambda z = 4 \)

### Step 2: Form the coefficient matrix
The coefficient matrix \( A \) for the system of equations is formed by the coefficients of \( x \), \( y \), and \( z \):
\[ A = \begin{bmatrix} 2 & -1 & 2 \\ 1 & -2 & 1 \\ 1 & 1 & \lambda \end{bmatrix} \]

### Step 3: Calculate the determinant of the coefficient matrix
To find the value of \( \lambda \) such that the system has no solution, we need to set the determinant of the matrix \( A \) to zero:
\[ \text{det}(A) = 0 \]

Calculating the determinant:
\[ \text{det}(A) = \begin{vmatrix} 2 & -1 & 2 \\ 1 & -2 & 1 \\ 1 & 1 & \lambda \end{vmatrix} \]

Using the rule of Sarrus or cofactor expansion, we can compute the determinant:
\[ = 2 \begin{vmatrix} -2 & 1 \\ 1 & \lambda \end{vmatrix} - (-1) \begin{vmatrix} 1 & 1 \\ 1 & \lambda \end{vmatrix} + 2 \begin{vmatrix} 1 & -2 \\ 1 & 1 \end{vmatrix} \]

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

Putting it all together:
\[ \text{det}(A) = 2(-2\lambda - 1) + (\lambda - 1) + 2(3) \]
\[ = -4\lambda - 2 + \lambda - 1 + 6 \]
\[ = -4\lambda + \lambda + 3 = -3\lambda + 3 \]

### Step 4: Set the determinant to zero
To find the value of \( \lambda \) such that the determinant is zero:
\[ -3\lambda + 3 = 0 \]
\[ -3\lambda = -3 \]
\[ \lambda = 1 \]

### Conclusion
The value of \( \lambda \) such that the given system of equations has no solution is:
\[ \lambda = 1 \]