Question

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

• A. $\pm 2$
• B. $\pm 3$
• C. $\pm 5$
• D. $\pm 0$

Answer 1

Answer: (B)

$\pm 3$

given, $\left|A^3\right|=125$

$\Rightarrow {\left|A\right|}^3=5^3$

$\Rightarrow \left|A\right|=5$

$\Rightarrow \left|\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right|=5$

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

$\Rightarrow {\alpha }^2=5+4=9$

$\therefore \alpha =\pm 3$