Question

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

We know $\left|A^n\right|={\left|A\right|}^n$
Since, $\left|A^3\right|=125\Rightarrow {\left|A\right|}^3=125$
$\Rightarrow \left|\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right|=5$

$\Rightarrow {\alpha }^2-4=5$

$\Rightarrow \alpha =\pm 3$