The system of linear equations x + y + z = 2 2x + 3y + 2z = 5 2x +...

Here, D = \left|\begin{cases}1 & 1 & 1 \\ 2 & 3 & 2 \\ 2 & 3 & {a}^{2} - 1\end{cases}\right|

$= {a}^{2} - 3$

Hence at $| a | = \sqrt{3}$ , $D = 0$ . So system don't have unique solution.

At $a = 4$ , $D \ne 0$ , So, system has unique solution. Hence not inconsistent.

At $| a | = \sqrt{3}, D = 0$ and ${2}^{nd}$ & ${3}^{rd}$ equation are parallel. Hence no solution are possible.

Hence, system is inconsistent for $| a | = \sqrt{3}$