The system of linear equations $x+y+z=2,2x+3y+2z=5,2x+3y+(a^2-1)z=a+1$ 

The given system of linear equation can be written as $\left[\begin{array}{ccc}1& 1& 1\\ 2& 3& 2\\ 2& 3& a^2-1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}2\\ 5\\ a+1\end{array}\right]$

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

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

${\Delta }_2=\left|\begin{array}{ccc}1& 2& 1\\ 2& 5& 2\\ 2& a+1& a^2-1\end{array}\right|=a^2-3$

${\Delta }_3=\left|\begin{array}{ccc}1& 1& 2\\ 2& 3& 5\\ 2& 3& a+1\end{array}\right|=a-4$

When $\left|a\right|=\sqrt{3},\Delta =0$ But, ${\Delta }_3=\pm \sqrt{3},\Delta =0$

System of linear equations is no solution when $\left|a\right|=\sqrt{3}$