MCQ
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
- A$\pm 1$
- B$\pm 2$
- C$\pm \sqrt{5}$
- ✓$\pm \sqrt{7}$
✓
Answer
Correct option: D.
$\pm \sqrt{7}$
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}$
$\left|A^3\right|=27$
$\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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