Matrices & Determinants
Infinite Solutions Condition
nta_pyq_2025_apr
Grade 12
Question:
If the system of equations $(\lambda-1)x + (\lambda-4)y + \lambda z = 5$, $\lambda x + (\lambda-1)y + (\lambda-4)z = 7$, $(\lambda+1)x + (\lambda+2)y - (\lambda+2)z = 9$ has infinitely many solutions, then $\lambda^2 + \lambda$ is equal to:
Step-by-Step Solution
Key Concept: Set $D = 0$ to find candidate values of $\lambda$, then verify using $D_x = 0$ to identify the valid $\lambda$.
$D = 0 \Rightarrow (\lambda-3)(2\lambda+1) = 0 \Rightarrow \lambda = 3$ or $\lambda = -1/2$. Checking $D_x = 0$: only $\lambda = 3$ satisfies it. $\lambda^2 + \lambda = 9 + 3 = 12$.
Correct Answer: 12