Question

Which of the following option(s) represents a possible value k, satisfying

$\left[ \begin{matrix} 1+\frac{1}{\sqrt{3}} & 1-\frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}}-1 & 1+\frac{1}{\sqrt{3}} \end{matrix} \right]^{2021} = (\frac{2}{3\sqrt{3}})^{\frac{6063-3k}{2}} \left[ \begin{matrix} 1-\frac{1}{\sqrt{3}} & 1+\frac{1}{\sqrt{3}} \\ -1-\frac{1}{\sqrt{3}} & -1+\frac{1}{\sqrt{3}} \end{matrix} \right]^{k}$?

• A. 12173
• B. 48413
• C. 12217
• D. 24265

Answer 1

Answer: (C)

12217

Answer 2

Here's how to solve the problem:

1. Complex Number Representation: Convert the 2x2 matrices into complex numbers. This simplifies the matrix operations into complex number operations.

   • Matrix A corresponds to the complex number a = (1 + 1/√3) + i(1 – 1/√3).
   • Matrix B corresponds to the complex number b = (1 – 1/√3) + i(1 + 1/√3).

2. Polar Form: Find the magnitude and argument (angle) of the complex numbers.

   • |a| = √(8/3) and arg(a) = 15°
   • |b| = √(8/3) and arg(b) = 75°

3. Apply the Power: Raise the complex numbers to the given powers.

   • A^2021 corresponds to a^2021, which has an angle of 2021 * 15°.
   • B^k corresponds to b^k, which has an angle of k * 75°.

4. Equate Arguments: For the matrix equality to hold (up to a positive real factor), the difference in angles must be an integer multiple of 360°.

   2021 * 15° – k * 75° = 360° * m, where m is an integer.

5. Solve for k: Rearrange the equation to solve for k in terms of m.

   k = (2021 – 24m) / 5

   Since k must be positive, let m = -M (where M is a positive integer):

   k = (2021 + 24M) / 5

6. Test the Options: Substitute each of the given options for k and see if you get an integer value for M.

   • For k = 12217:

     2021 + 24M = 5 * 12217 = 61085 --> 24M = 59064 --> M = 2461 (an integer)

   The other options do not yield integer values for M.

Therefore, the correct answer is k = 12217.