Question

If $A=\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]$ and $|A^3|=27$, then $\alpha =$ ____

• A. $\pm 1$
• B. $\pm 2$
• C. $\pm \sqrt{7}$
• D. $\pm \sqrt{5}$

Answer 1

Answer: (C)

$\pm \sqrt{7}$

Using fundamental theorem of determinant $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$

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

It is also given that $|A^3|=27$

$\Rightarrow |A{|}^3=27$, basic law $|A^n|=|A{|}^n$

$\Rightarrow ({\alpha }^2-4{)}^3=27=3^3$

$\Rightarrow {\alpha }^2-4=3$, take cube root booth sides

$\Rightarrow {\alpha }^2=7$

$\therefore \alpha =\pm \sqrt{7}$