Question

Let $\lambda \in R$. The system of linear equations
$2 x_{1}-4 x_{2}+\lambda x_{3}=1$
$x_{1}-6 x_{2}+x_{3}=2$
$\lambda x_{1}-10 x_{2}+4 x_{3}=3$
is inconsistent for :

• A. exactly one negative value of $\lambda$
• B. exactly one positive value of $\lambda$
• C. every value of $\lambda$
• D. exactly two values of $\lambda$

Answer 1

Answer: (A)

$D=\begin{vmatrix}2&-4&\lambda\\ 1&-6&1\\ \lambda&-10&4\end{vmatrix}$
$=2(3 \lambda+2)(\lambda-3)$
$D _{1}=-2(\lambda-3)$
$D _{2}=-2(\lambda+1)(\lambda-3)$
$D _{3}=-2(\lambda-3)$
When $\lambda=3$, then
$D = D _{1}= D _{2}= D _{3}=0$
$\Rightarrow $ Infinite many solution
when $\lambda=-\frac{2}{3}$ then $D _{1}, D _{2}, D _{3}$ none of them is
zero so equations are inconsistant
$\therefore \lambda=-\frac{2}{3}$