Question:
If $A=\left[\begin{array}{cc}\alpha & 2 \\ 2 & \alpha\end{array}\right]$ and $\left|A^{3}\right|=125$, then $\alpha=$
Solution:
Given:
$A=\left[\begin{array}{ll}\alpha & 2 \\ 2 & \alpha\end{array}\right]$
$\left|A^{3}\right|=125$
Now,
$\left|A^{3}\right|=125$
$\Rightarrow|A|^{3}=5^{3}$ $\left(\because\left|A^{n}\right|=|A|^{n}\right)$
$\Rightarrow|A|=5$
$\Rightarrow\left|\begin{array}{ll}\alpha & 2 \\ 2 & \alpha\end{array}\right|=5$
$\Rightarrow \alpha^{2}-4=5$
$\Rightarrow \alpha^{2}=5+4$
$\Rightarrow \alpha^{2}=9$
$\Rightarrow \alpha=\pm 3$
Hence, $\alpha=\pm \underline{3}$.