Question

Given \[ 2x - y + 2z = 2, \quad x - 2y + z = -4, \quad x + y + \lambda z = 4, \] then the value of  \( \lambda \) such that the given system of equations has no solution is:

• A. -3
• B. 1
• C. 0
• D. 3

Hint:

In linear systems, to determine when a system has no solution, compute the determinant of the coefficient matrix. If the determinant is zero, the system may be inconsistent.

Answer 1

The Correct Option is D

\[ \begin{vmatrix} 2 & -1 & 2 \\ 1 & -2 & -1 \\ 1 & 1 & \lambda \end{vmatrix} = 0 \] This leads to the calculation: \[ = 2(-2\lambda + 1) + 1(\lambda + 1) + 2(C) = 0 \] \[ \Rightarrow -4\lambda + 2 + \lambda + 1 + 6 = 0 \] \[ \Rightarrow -3\lambda + 9 = 0 \] \[ \Rightarrow \lambda = 3 \]
So, the correct answer is Option D.