The system of linear equations $x+y+z=2,2x+3y+2z=5,2x+3y+\left({\alpha }^2-1\right)z=\alpha +1$

$$\begin{gathered} D=\left|\begin{array}{ccc}1& 1& 1\\ 2& 3& 2\\ 2& 3& a^2-1\end{array}\right| \\ {C}_2\to C_2-C_1\text{ and }{C}_3\to C_3-C_1 \\ D=\left|\begin{array}{ccc}1& 1& 0\\ 2& 1& 0\\ 2& 1& a^2-3\end{array}\right|=\left(a^2-3\right) \end{gathered}$$

since if $a^2\ne 3$then only it will have unique solution. So at a = 4 unique solution exist.