$\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| =$

For real numbers $\alpha$ and $\beta$, consider the following system of linear equations:

$x+y-z=2, x+2 y+\alpha z=1,2 x-y+z=\beta$. If the system has infinite solutions, then $\alpha+\beta$ is equal to $.....$

The values of $x$ in the following determinant equation, $\left| {\,\begin{array}{*{20}{c}}{a + x}&{a - x}&{a - x}\\{a - x}&{a + x}&{a - x}\\{a - x}&{a - x}&{a + x}\end{array}\,} \right| = 0$ are

The roots of the equation $\left| {\,\begin{array}{*{20}{c}}{1 + x}&1&1\\1&{1 + x}&1\\1&1&{1 + x}\end{array}\,} \right| = 0$   are

The system of equations ${x_1} - {x_2} + {x_3} = 2,$ $\,3{x_1} - {x_2} + 2{x_3} = - 6$ and $3{x_1} + {x_2} + {x_3} = - 18$ has