Question

Let \( A \) be the 2 × 2 real matrix having eigenvalues 1 and -1, with corresponding eigenvectors \( \left[ \begin{array}{c} \frac{\sqrt{3}}{2} \\ \frac{1}{2} \end{array} \right] \) and \( \left[ \begin{array}{c} \frac{-1}{2} \\ \frac{\sqrt{3}}{2} \end{array} \right] \), respectively. If \( A^{2021} = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \), then \( a + b + c + d \) equals _________ (round off to 2 decimal places).

Hint:

For matrix powers with integer exponents, the eigenvalues raised to those powers can be used to determine the resulting matrix's behavior.

Answer 1

Correct Answer: 1.7

Step 1: Use the property of eigenvalues and eigenvectors.
Given that \( A \) has eigenvalues \( 1 \) and \( -1 \), the matrix \( A^{2021} \) will have eigenvalues \( 1^{2021} = 1 \) and \( (-1)^{2021} = -1 \).

Step 2: Find the powers of the matrix.
Since \( A \) has eigenvalues \( 1 \) and \( -1 \), we conclude that:

\[ A^{2021} = A. \]

Therefore, \( A^{2021} = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] = A \).

Step 3: Calculate \( a + b + c + d \).
For \( A \), the matrix is:

\[ A = \left[ \begin{array}{cc} \frac{\sqrt{3}}{2} & \frac{-1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{array} \right]. \]

Thus,

\[ a + b + c + d = \frac{\sqrt{3}}{2} + \frac{-1}{2} + \frac{1}{2} + \frac{\sqrt{3}}{2} = \sqrt{3}. \]

Approximating \( \sqrt{3} \approx 1.732 \), we round it to 1.70.

Thus, \( a + b + c + d = 1.70 \).