Let the system of linear equations x + 2y + z = 2 , + 3y - z = α, -...

Given, the system of linear equations $x + 2y + z = 2$, $\alphax + 3y - z = \alpha,$ and $- \alphax + y + 2z = - \alpha$

Now for system to be inconsistent, $∆ = 0 \text{ and } \text{either of}{∆}_{1}, {∆}_{2}\text{or}{∆}_{3} \ne 0$

So, \Delta = \left|\begin{cases}1 & 2 & 1 \\ 2 & 3 & - 1 \\ - 2 & 1 & 2\end{cases}\right|

$= \left(6 + y\right) - 2\left(2\alpha - \alpha\right) + 1\left(\alpha + 3\alpha\right)$

$= 7 - 2\alpha + 4\alpha$

$= 7 + 2\alpha$

$\Delta = 0 ⇒ \alpha = - \dfrac{7}{2}$

Now calculating, {\Delta}_{1} = \left|\begin{cases}2 & 2 & 1 \\ \alpha & 3 & - 1 \\ - \alpha & 1 & 2\end{cases}\right|

$= 14 + 2\alpha$

Where $\alpha = - \dfrac{7}{2}$ so, ${∆}_{1} = 14 - 7 = 7$

${\Delta}_{1} \ne 0$, hence system is inconsistent at $\alpha = \dfrac{- 7}{2}$