The system of equations $ \begin{aligned} & \alpha x+y+z=\alpha-1 \ & x+\alpha y+z=\alpha-1 \ & x+y+\alpha z=\alpha-1 \end{aligned} $ has no solution, if $\alpha$ is

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Aicharya · Accepted Answer

$ \begin{aligned} & \alpha x+y+z=\alpha-1 \ & x+\alpha y+z=\alpha-1 \ & x+y+z \alpha=\alpha-1 \ & \Delta=\left|\begin{array}{ccc} \alpha & 1 & 1 \ 1 & \alpha & 1 \ 1 & 1 & \alpha \end{array}ight| \ & =\alpha\left(\alpha^2-1ight)-1(\alpha-1)+1(1-\alpha) \ & =\alpha(\alpha-1)(\alpha+1)-1(\alpha-1)-1(\alpha-1) \ & \Rightarrow(\alpha-1)\left[\alpha^2+\alpha-1-1ight]=0 \ & \Rightarrow(\alpha-1)\left[\alpha^2+\alpha-2ight]=0 \ & {\left[\alpha^2+2 \alpha-\alpha-2ight]=0} \ & (\alpha-1)[\alpha(\alpha+2)-1(\alpha+2)]=0 \ & (\alpha-1)=0, \alpha+2=0 \Rightarrow \alpha=-2,1 ; 	ext { but } \alpha 
eq 1 \end{aligned} $

The system of equations $ \begin{aligned} & \alpha x+y+z=\alpha-1 \\ & x+\alpha y+z=\alpha-1 \\ & x+y+\alpha…

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