Let λ ∈ R . The system of linear equations 2 x _ 1 - 4 x _ 2 + λ x...

Here, we will use Cramer's Rule according to which, $D = 0$ and, ${D}_{1} \ne 0$ if the system of linear equations is inconsistent. D = \left|\begin{cases}2 & - 4 & \lambda \\ 1 & - 6 & 1 \\ \lambda & - 10 & 4\end{cases}\right| = 2\left(- 24 + 10\right) + 4\left(4 - \lambda\right) + \lambda\left(- 10 + 6\lambda\right)

$= 2\left(3{\lambda}^{2} - 7\lambda - 6\right) = 2\left(\lambda - 3\right)\left(3\lambda + 2\right)$

$D = 0 ⇒ \lambda = 3, - \dfrac{2}{3}$

{D}_{1} = \left|\begin{cases}1 & - 4 & \lambda \\ 2 & - 6 & 1 \\ 3 & - 10 & 4\end{cases}\right| = 1\left(- 24 + 10\right) + 4\left(8 - 3\right) + \lambda\left(- 20 + 18\right) = 2\left(3 - \lambda\right)

For $\lambda = - \dfrac{2}{3}, {D}_{1} \ne 0$