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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
exactly one negative value of $\lambda$.
exactly one positive value of $\lambda$.
every value of $\lambda$.
exactly two values of $\lambda$.
Solution
$D=\left|\begin{array}{ccc}2 & -4 & \lambda \\ 1 & -6 & 1 \\ \lambda & -10 & 4\end{array}\right|$
$=2(3 \lambda+2)(\lambda-3)$
$D_{1}=-2(\lambda-3)$
$D _{2}=-2(\lambda+1)(\lambda-3)$
$D_{3}=-2(\lambda-3)$
When $\overline{\lambda=3}$, then
$D = D _{1}= D _{2}= D _{3}=0$
$\Rightarrow$ Infinite many solution
when $\left|\lambda=-\frac{2}{3}\right|$ then $D _{1}, D _{2}, D _{3}$ none of them
is zero so equations are inconsistant
$\therefore \lambda=-\frac{2}{3}$
