Question

If the system of linear equations
$2 x+3 y-z=-2$
$x+y+z=4$
$x-y+|\lambda| z=4 \lambda-4$
where $\lambda \in R$, has no solution, then

• A. $\lambda=7$
• B. $\lambda=-7$
• C. $\lambda=8$
• D. $\lambda^{2}=1$

Answer 1

Answer: (B)

$\begin{vmatrix} 2 & 3 & -1 \\ 1 & 1 & 1 \\ 1 & -1 & |\lambda|
\end{vmatrix}=0$
$\Rightarrow|\lambda|=7 \Rightarrow \lambda=\pm 7$ ...(1)
System :
$2 x+3 y-z=-2$ ...(2)
$x+ y+ z=4$ ...(3)
$x-y+|\lambda| z=4 \lambda-4$ ...(4)
Eliminating y from equal (2) & (3) we get
$x+4 z=14$ ...(5)
$(3)+(4) \Rightarrow x+\left(\frac{|\lambda|+1}{2}\right) z=2 \lambda$ ...(6)
Clearly for $\lambda=-7$, system is inconsistent.