The value of `lambda`, such that the following system of equations has no solution, is
x - 2y + z = - 4
2x - y - 2z = 2
x + y + `lambda`z = 4

Options

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

Answer

To find the value of \( \lambda \) such that the following system of equations has no solution:

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

we will set up the augmented matrix and find the determinant. The system of equations will have no solution if the determinant of the coefficient matrix is zero.

### Step 1: Write the coefficient matrix

The coefficient matrix \( A \) for the given system of equations is:

\[ A = \begin{bmatrix} 1 & -2 & 1 \\ 2 & -1 & -2 \\ 1 & 1 & \lambda \end{bmatrix} \]

### Step 2: Calculate the determinant of the matrix

To find the determinant of matrix \( A \), we can use the formula for the determinant of a 3x3 matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:

\[ \text{det}(A) = 1 \cdot \left((-1) \cdot \lambda - (-2) \cdot 1\right) - (-2) \cdot \left(2 \cdot \lambda - (-2) \cdot 1\right) + 1 \cdot \left(2 \cdot 1 - (-1) \cdot 1\right) \]

### Step 3: Simplify the determinant expression

Calculating each part:

1. First term:
\[ 1 \cdot (-\lambda + 2) = -\lambda + 2 \]

2. Second term:
\[ -(-2) \cdot (2\lambda + 2) = 2(2\lambda + 2) = 4\lambda + 4 \]

3. Third term:
\[ 1 \cdot (2 + 1) = 3 \]

Putting it all together:

\[ \text{det}(A) = -\lambda + 2 + 4\lambda + 4 + 3 = 3\lambda + 9 \]

### Step 4: Set the determinant to zero

For the system to have no solution, we set the determinant equal to zero:

\[ 3\lambda + 9 = 0 \]

### Step 5: Solve for \( \lambda \)

Now, solve for \( \lambda \):

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

### Conclusion

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

\[ \lambda = -3 \]