The value of $x,$ if $\left| {\,\begin{array}{*{20}{c}}{ - x}&1&0\\1&{ - x}&1\\0&1&{ - x}\end{array}\,} \right| = 0 $ is equal to

The existance of the unique solution of the system of equations$2x + y + z = \beta $ , $10x - y + \alpha z = 10$ and $4x+ 3y-z =6$ depends on

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 $.....$

If $a, b, c$ are non-zero real numbers and if the system of equations $(a - 1 )x = y + z,$  $(b - 1 )y = z + x ,$ $(c - 1 )z= x + y,$ has a non-trivial solution, then $ab + bc + ca$ equals

$\left| {\,\begin{array}{*{20}{c}}{1/a}&1&{bc}\\{1/b}&1&{ca}\\{1/c}&1&{ab}\end{array}\,} \right| = $

If the system of equations $x + ay = 0,$ $az + y = 0$ and $ax + z = 0$ has infinite solutions, then the value of $a$ is