Question

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:

• A. 5
• B. 3
• C. 9
• D. 17

Answer 1

Answer: (A)

5

Answer 2

Step 1: Calculate the Determinant of A

$|A| = \alpha^2 - \beta^2$

Step 2: Use the Condition $|2A|^3 = 2^{21}$

We know that:

$|2A| = 2^3 |A| = 2^{21} \Rightarrow |A| = 2^4 = 16$

Step 3: Set Up the Equation

$\alpha^2 - \beta^2 = 16$

Factor as $(\alpha + \beta)(\alpha - \beta) = 16$.

Step 4: Solve for Possible Values of $\alpha$

Possible integer solutions for $(\alpha, \beta)$ that satisfy the equation give $\alpha = 4$ or $\alpha = 5$.

Since $\alpha = 5$ satisfies the condition, we choose $\alpha = 5$.

So, the correct answer is: 5