If $A=\left[\begin{array}{ccc}1& 1& 1\\ 2& 1& -3\\ -1& 2& 3\end{array}\right]$, then co-factors of $3^{rd}$ row are

If $a, b, c$ are nonzero, then the system of equations, $(\alpha +a)x+\alpha y+\alpha z=0$, $\alpha x+(\alpha +b)y+\alpha z=0$ and $\alpha x+\alpha y+(\alpha +c)z=0$ has a non-trivial solution, if

The system of homogeneous equations:

$tx+\left(t+1\right)y+\left(t-1\right)z=0,$

$\left(t+1\right)x+ty+\left(t+2\right)z=0,$

$\left(t-1\right)x+\left(t+2\right)y+tz=0$ has a non-trivial solution for