Question

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

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

Answer 1

Answer: (C)

$\pm 3$

Given that: $A=\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]$

Then, $|A^3|=|A{|}^3$

$\therefore (\alpha {)}^2-4={125}^{\frac{1}{3}}=5$

$\therefore (\alpha {)}^2=9$

$\therefore \alpha =\pm 3$