For a real number , if the system [ ccc 1 & & ^2 & 1 & ^2 & & 1 ]… - Vidyadip

Question

For a real number $\alpha$, if the system$\left[\begin{array}{ccc}1 & \alpha & \alpha^2 \\ \alpha & 1 & \alpha \\ \alpha^2 & \alpha & 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}1 \\ -1 \\ 1\end{array}\right]$ of linear equations, has infinitely many solutions, then $1+\alpha+\alpha^2=$

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Answer

Here, $D=\left|\begin{array}{lll}1 & \alpha & \alpha^2 \\ \alpha & 1 & \alpha \\ \alpha^2 & \alpha & 1\end{array}\right|=0$
which gives $\alpha=-1$ or $+1$
For $\alpha=1,$ the equations become
$ x+y+z=1$
$x+y+z=-1$
and $x+y+z=1$
which give no solution
For $\alpha=-1,$ the equations become
$x-y+z=1$
$-x+y-z=-1$
$x-y+z=1$
which are all same and hence infinitely many solutions
Hence, $\alpha=-1$
$\Rightarrow 1+\alpha+\alpha^2=1$

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