If A= [ ll & 2 2 & ] and |A^3 |=27, then the value of is - Vidyadip

Question

If $A=\left[\begin{array}{ll}\alpha & 2 \\ 2 & \alpha\end{array}\right]$ and $\left|A^3\right|=27$, then the value of $\alpha$ is

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Answer

Given, $A=\left[\begin{array}{ll}\alpha & 2 \\ 2 & \alpha\end{array}\right]$
$ \left|A^3\right|=27$
$ \Rightarrow|A|^3=27\left[\because\left|A^n\right|=|A|^n\right] $
$\Rightarrow|A|=3$
From $(i)$ and $(ii),$ we get
$\Rightarrow \alpha^2-4=3 $
$\Rightarrow \alpha^2=7 $
$\Rightarrow \alpha= \pm \sqrt{7}$

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