The system of linear equations $x+y+z=2,2x+3y+2z=52x+3y+\left(a^2-1\right)z=a+1$

According to cramer’s rule , here 

$D=\left|\begin{array}{ccc}1& 1& 1\\ 2& 3& 2\\ 2& 3& a^2-1\end{array}\right|=\left|\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 2& 1& a^2-3\end{array}\right|$

$\text{ ( applying }\left.C_2\to C_2-C_1\text{ and }{C}_3\to C_3-C_1\right)$

$=a^2-3$         ( Expanding along  $R_1$) 

$\text{ And }{D}_1=\left|\begin{array}{ccc}2& 1& 1\\ 5& 3& 2\\ a+1& 3& a^2-1\end{array}\right|=\left|\begin{array}{ccc}2& 1& 0\\ 5& 3& -1\\ a+1& 3& a^2-1-3\end{array}\right|$

$\text{ ( Applving }\left.C_2\to C_2-C_2\right)$

$=\left|\begin{array}{ccc}2& 0& 0\\ 5& 3-\frac{5}{2}& -1\\ a+1& 3-\frac{\left(a+1\right)}{2}& a^2-1-3\end{array}\right|$

$\text{ ( Applving }\left.C_2\to C_2-C_2\right)$

$=\left|\begin{array}{ccc}2& 0& 0\\ 5& \frac{1}{2}& -1\\ a+1& \frac{5}{2}-\frac{a}{2}& a^2-4\end{array}\right|$

$=2\left[\frac{1}{2}\left(a^2-4\right)+\left(\frac{5}{2}-\frac{a}{2}\right)\right]\left[\text{ expanding along }{R}_1\right]$

$=2\left[\frac{a^2}{2}-2+\frac{5}{2}-\frac{a}{2}\right]=a^2-4+5-a=a^2-a+1$

$\text{ Clearly, when }a=4\text{, then }D=13\ne 0\Rightarrow \text{ unique solution and }$

$\text{ When }\left|a\right|=\sqrt{3}\text{, then }\mathrm{D}=0\text{ and }{D}_1\ne 0$

$\therefore \text{ when }\left|a\right|=\sqrt{3}\text{, then the system has no solution i.e. system is inconsistent. }$