Question

for no solution $\Delta = 0$

$${\left| \begin{matrix} 3 & 1 & 4 \\ 2 & \alpha & - 1 \\ 1 & 2 & 1 \end{matrix} \right| = 0 }{\Rightarrow 3(a + 2) + 1( - 1 - 2) + 4(4 - \alpha) = 0 }{\Rightarrow 19 - \alpha = 0 \Rightarrow \alpha = 19 }$$& for $\alpha = 19$

$${\Delta_{x} = \left| \begin{matrix} 3 & 1 & 4 \\ - 3 & 19 & - 1 \\ 4 & 2 & 1 \end{matrix} \right| = 3(21) + 1( - 1) + 4( - 82) }{\neq 0 }$$∴ no solution for $\alpha = 19$