For the system of linear equations $\alpha x+y+z=1,x+\alpha y+z=1,x+y+\alpha z=\beta$, which one of the following statements is NOT correct?

Question

For the system of linear equations $\alpha x+y+z=1,x+\alpha y+z=1,x+y+\alpha z=\beta$, which one of the following statements is NOT correct?

It has infinitely many solutions if $\alpha=1$ and $\beta=1$

It has infinitely many solutions if $\alpha=2$ and $\beta=-1$

$x+y+z=\frac{3}{4}$ if $\alpha=2$ and $\beta=1$

It has no solution if $\alpha=-2$ and $\beta=1$

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