If for the matrix $A = \begin{bmatrix} \alpha & -2 \\ -2 & \alpha \end{bmatrix}, |A^3| = 125$, then the value of $\alpha$ is:

$\pm 3$

The correct answer is Option (1) → $\pm 3$ ##

Given, $A = \begin{bmatrix} \alpha & -2 \\ -2 & \alpha \end{bmatrix}$

$⇒|A| = \alpha^2 - 4$ ...(i)

Also, given $|A^3| = 125$

$⇒|A|^3 = 125$

$⇒|A| = 5$

From eq. (i): $\alpha^2 - 4 = 5$

$⇒ \alpha^2 = 9$

$⇒ \alpha = \pm 3$