Para que valor de $\alpha$ o sistema

$\{\begin{array}{l}2x-y-z=1\\ x+2y+3z=0\\ -x-y+\alpha z=0\end{array}$

tem solução única $(x,y,z)$ dada por:

$x=\det \left(\begin{array}{ccc}1& -1& -1\\ 0& 2& 3\\ 0& -1& \alpha \end{array}\right)$

$y=\det \left(\begin{array}{ccc}2& 1& -1\\ 1& 0& 3\\ -1& 0& \alpha \end{array}\right)$

$z=\det \left(\begin{array}{ccc}2& -1& 1\\ 1& 2& 0\\ -1& -1& 0\end{array}\right)$ ?