Determinants
General
Grade 12
Question:
If the system of equations $x + \lambda y + 1 = 0, \lambda x + y + 1 = 0$ & $x + y + \lambda = 0$ is consistent, then find the value of $\lambda$.
Step-by-Step Solution
Key Concept: General
For consistency of the given system of equations<br>$$D = \begin{vmatrix} 1 & \lambda & 1 \\ \lambda & 1 & 1 \\ 1 & 1 & \lambda \end{vmatrix} = 0$$<br>$\Rightarrow 3\lambda = 1 + 1 + \lambda^3$ or $\lambda^3 - 3\lambda + 2 = 0$<br>$\Rightarrow (\lambda - 1)^2 (\lambda + 2) = 0$<br>$\Rightarrow \lambda = 1$ or $\lambda = -2$
Correct Answer: A