Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix}$ and $|2A|^3 = 2^{21}$ where $\alpha, \beta \in \mathbb{Z}$, Then a value of $\alpha$ is
Solution
Given: $A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix}$ and $|2A|^3=2^{21}$
Now, $|2A|^3=2^{21}$
$\Rightarrow |2A|=2^7$
$\Rightarrow 2^3|A|=2^7$
$\Rightarrow |A|=16$
$\Rightarrow \alpha^2-\beta^2=16$
Now, taking value from the option we get,
$\alpha=5$ as $\alpha, \beta \in \mathbb{Z}$, so for other value of $\alpha$, $\beta$ is not an integer.