This is a question which was asked in a high-school exam held in India(JEE ADVANCED). Going by Cramer's rule, for infinite solutions, I should get $D=D_1=D_2=D_3=0$ (where $D$ is the original determinant and $D_1, D_2, D_3$ are the respective coefficient determinants). Using these, I arrive at $\alpha^2$=1, so that $\alpha$=1,-1. But, $\alpha=1$ yields no solution here (if I write down the system of equations using $\alpha=1$). Why does it happen so? Is this a rare failure of Cramer's rule? How should we explain this unexpected result?
For Cramer's rule for 3 equations, if $D\ne 0$, then you have a single solution. IF $D=0$ and any one of $D_1, D_2, D_3$ is non-zero, then the system is inconsistent. If all of those are zero, you still have two possibilities, one where the system is indeterminate and one where the system is inconsistent.