Question

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

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

Answer 1

The correct option is C

$\pm 3$

Explanation for the correct otpion:

Finding the value of $\mathit{\alpha }$:

Given,

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

Now,

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

$\begin{array}{rcl}\left|A^3\right|& =& 125\\ \left|A\right|& =& 5\end{array}$

Therefore,

$$\begin{gathered} 5={\alpha }^2-4 \\ 9={\alpha }^2 \\ \alpha =\pm 3 \end{gathered}$$

Hence, correct option is $\mathbf{\left(}\mathit{C}\mathbf{\right)}$