Question

The value of $\left|\alpha \right|$ for which the system of equation $\alpha x+y+z=\alpha -1,x+\alpha y+z=\alpha -1,x+y+\alpha z=\alpha -1$ has no solution, is ..............

Answer 1

Augumented matrix is $\left[\begin{array}{cccc}\alpha & 1& 1& \alpha -1\\ 1& \alpha & 1& \alpha -1\\ 1& 1& \alpha & \alpha -1\end{array}\right]$
given system of equations has no solution.
$\Rightarrow \Delta =0$ and ${\Delta }_x\ne 0$
${\Delta }_x=\left|\begin{array}{ccc}\alpha -1& 1& 1\\ \alpha -1& \alpha & 1\\ \alpha -1& 1& \alpha \end{array}\right|=(\alpha -1{)}^3\ne 0$
$\Rightarrow \alpha \ne 1$
$\Delta =\left|\begin{array}{ccc}\alpha & 1& 1\\ 1& \alpha & 1\\ 1& 1& \alpha \end{array}\right|=(\alpha +2)(\alpha -1{)}^2=0$
$\Rightarrow \alpha =-2$ ($\because \alpha \ne 1$)
$\therefore |\alpha |=2$