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If $\alpha$ and $\beta$ are the roots of the equation $1+\mathrm{x}+\mathrm{x}^{2}=0$, then
the matrix product $\left[\begin{array}{cc}1 & \beta \\ \alpha & \alpha\end{array}\right]\left[\begin{array}{cc}\alpha & \beta \\ 1 & \beta\end{array}\right]$ is equal to
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the matrix product $\left[\begin{array}{cc}1 & \beta \\ \alpha & \alpha\end{array}\right]\left[\begin{array}{cc}\alpha & \beta \\ 1 & \beta\end{array}\right]$ is equal to
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Verified Answer
The correct answer is:
$\left[\begin{array}{cc}-1 & -1 \\ -1 & 2\end{array}\right]$
$\alpha, \beta$ are roots of the equation $1+x+x^{2}=0$. $1+x+x^{2}=0 \Rightarrow x^{2}+x+1=0$
Solving for $x, x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}=\frac{-1 \pm \sqrt{1^{2}-4(1)(1)}}{2(1)}$
$=\frac{-1 \pm \sqrt{-3}}{2}=\frac{-1 \pm \sqrt{3} \mathrm{i}}{2}$
$\therefore$ roots are $\frac{-1 \pm \sqrt{3} \mathrm{i}}{2}$ and $\frac{-1-\sqrt{3} \mathrm{i}}{2}$
i.e., $\alpha=\mathrm{a}, \beta=\omega^{2}$
$\left[\begin{array}{ll}1 & \beta \\ \alpha & \alpha\end{array}\right]\left[\begin{array}{ll}\alpha & \beta \\ 1 & \beta\end{array}\right]=\left[\begin{array}{cc}\alpha+\beta & \beta+\beta^{2} \\ \alpha^{2}+\alpha & \alpha \beta+\alpha \beta\end{array}\right]$
$=\left[\begin{array}{ll}\omega+\omega^{2} & \omega^{2}+\omega^{4} \\ \omega^{2}+\omega & \omega^{3}+\omega^{3}\end{array}\right]$
$=\left[\begin{array}{cc}-1 & \omega^{2}+\omega \\ -1 & 2 \omega^{3}\end{array}\right]$
$=\left[\begin{array}{cc}-1 & -1 \\ -1 & 2\end{array}\right]$
Solving for $x, x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}=\frac{-1 \pm \sqrt{1^{2}-4(1)(1)}}{2(1)}$
$=\frac{-1 \pm \sqrt{-3}}{2}=\frac{-1 \pm \sqrt{3} \mathrm{i}}{2}$
$\therefore$ roots are $\frac{-1 \pm \sqrt{3} \mathrm{i}}{2}$ and $\frac{-1-\sqrt{3} \mathrm{i}}{2}$
i.e., $\alpha=\mathrm{a}, \beta=\omega^{2}$
$\left[\begin{array}{ll}1 & \beta \\ \alpha & \alpha\end{array}\right]\left[\begin{array}{ll}\alpha & \beta \\ 1 & \beta\end{array}\right]=\left[\begin{array}{cc}\alpha+\beta & \beta+\beta^{2} \\ \alpha^{2}+\alpha & \alpha \beta+\alpha \beta\end{array}\right]$
$=\left[\begin{array}{ll}\omega+\omega^{2} & \omega^{2}+\omega^{4} \\ \omega^{2}+\omega & \omega^{3}+\omega^{3}\end{array}\right]$
$=\left[\begin{array}{cc}-1 & \omega^{2}+\omega \\ -1 & 2 \omega^{3}\end{array}\right]$
$=\left[\begin{array}{cc}-1 & -1 \\ -1 & 2\end{array}\right]$
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