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If \( \mathrm{A}=\left[\begin{array}{ll}\alpha & 2 \\ 2 & \alpha\end{array}\right] \) and \( \left|\mathrm{A}^{3}\right|=27 \), then \( \alpha= \)
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Verified Answer
The correct answer is:
\( \pm \sqrt{7} \)
Given that $A=\left[\begin{array}{ll}\alpha & 2 \\ 2 & \alpha\end{array}\right]$
And $\left|A^{3}\right|=27=|A|^{3}$
$\Rightarrow|A|=3$
So, $|A|=\alpha^{2}-4=3$
$\Rightarrow \alpha^{2}=7 \Rightarrow \alpha=\pm \sqrt{7}$
And $\left|A^{3}\right|=27=|A|^{3}$
$\Rightarrow|A|=3$
So, $|A|=\alpha^{2}-4=3$
$\Rightarrow \alpha^{2}=7 \Rightarrow \alpha=\pm \sqrt{7}$
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