Question

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

• A. $\pm$ 1
• B. $\pm$ 2
• C. $\pm$ 3
• D. $\pm$ 5

Answer 1

The correct option is C $\pm$ 3

Given, $A=\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]$
$\Rightarrow |A|={\alpha }^2-4$
Also, given $|A^3|=125$
$\Rightarrow |A{|}^3=125$
$\Rightarrow |A|=5$
$\Rightarrow {\alpha }^2-4=5$
$\Rightarrow {\alpha }^2=9$
$\Rightarrow \alpha =\pm 3$