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

To solve the given problem, we start with the matrix \( A \) and the equation \(|2A|^3 = 2^{21}\).

The matrix \( A \) is: 

1	0	0
0	\(\alpha\)	\(\beta\)
0	\(\beta\)	\(\alpha\)

The determinant of a block matrix or partitioned matrix \(|A|\) can be found using cofactor expansion or specific rules for symmetric matrices.

The determinant of \( A \) can be calculated as follows:

• Using the formula for the determinant of a block matrix: \(|A| = \begin{vmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{vmatrix}\)
• Applying cofactor expansion along the first row, we find: \(|A| = 1 \cdot \begin{vmatrix} \alpha & \beta \\ \beta & \alpha \end{vmatrix}\)
• The determinant of the \(2 \times 2\) submatrix is: \(= \alpha^2 - \beta^2\)
• Therefore, \(|A| = \alpha^2 - \beta^2\).

Substituting in \(|2A|\):

• Since every element of \( A \) is multiplied by 2, except for the zero rows and columns, \(|2A| = 2^3 \cdot |A| = 8(\alpha^2 - \beta^2)\).

Given that \(|2A|^3 = 2^{21}\), substitute to find:

• \((8(\alpha^2 - \beta^2))^3 = 2^{21}\)
• Which simplifies to: \(2^9 (\alpha^2 - \beta^2)^3 = 2^{21}\)
• By comparing exponents: \((\alpha^2 - \beta^2)^3 = 2^{12}\) or \((\alpha^2 - \beta^2) = 2^4 = 16\).

Thus, \(\alpha^2 - \beta^2 = 16\).

Assuming integers solutions with the smallest values for simplicity, consider:

• \(\alpha - \beta = 4\) and \(\alpha + \beta = -4\) can be solved simultaneously.
• Adding and subtracting these equations:
  • Adding: \(2\alpha = 8, \, \alpha = 4\)
  • Subtracting: \(2\beta = 0, \, \beta = 0\)

Thus, a possible solution is \(\alpha = 5\) when generalizing all options possible using similar calculations. The option consistent with provided choices is:

Correct Answer: 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