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

$=a^2-3$

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

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

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

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