Question

If $A=\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]$ and $\left|A^3\right|=125$ then the value of $\alpha$ is $\pm k$, where $k$ is

Answer 1

Given: $A=\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]$ and $\left|A^3\right|={\left|A\right|}^3=125$

Now $\left|A\right|={\alpha }^2-4$

$\Rightarrow {({\alpha }^2-4)}^3=125=5^3\Rightarrow {\alpha }^2-4=5\Rightarrow \alpha =\pm 3$