Consider the following system of equations $\alpha x+2y+z=1$ $2\alpha x+3y+z=1$ $3x+\alpha y+2z=\beta$ for some $\alpha,\beta\in \mathbb{R}$. Then which of the following is NOT correct.

Question

Consider the following system of equations

$\alpha x+2y+z=1$

$2\alpha x+3y+z=1$

$3x+\alpha y+2z=\beta$

for some $\alpha,\beta\in \mathbb{R}$. Then which of the following is NOT correct.

It has a solution for all $\alpha\ne-1$ and $\beta=2$

It has no solution if $\alpha=-1$ and $\beta\ne2$

It has no solution for $\alpha=-1$ and for all $\beta \in \mathbb{R}$

It has no solution for $\alpha=3$ and for all $\beta\ne2$

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