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

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
$\pm \sqrt{5}$
D
$\pm \sqrt{7}$

✓

Answer

$\begin{array}{l}
\text { 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
\end{array}
$
From (i) and (ii), we get
$
\Rightarrow \alpha^2-4=3 \Rightarrow \alpha^2=7 \Rightarrow \alpha= \pm \sqrt{7}
$

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