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

The Correct Option is A

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