Question

Let \( A \) be a \( 3 \times 3 \) matrix defined as:

\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & -1 \end{bmatrix} \]

Find the eigenvalues of \( A^{13} \).

• A. Eigenvalues of \( A^{13} \) are powers of the eigenvalues of \( A \).
• B. Eigenvalues of \( A^{13} \) are the eigenvalues of \( A \) raised to the power of 13.
• C. Eigenvalues of \( A^{13} \) are the eigenvalues of \( A \) raised to the power of 13.
• D. Eigenvalues of \( A^{13} \) are the eigenvalues of \( A \) raised to the power of 13.

Hint:

<div>Eigenvalues of Matrix Powers: The eigenvalues of A\textsuperscript{n} are the eigenvalues of A raised to the power n.</div>

Answer 1

The Correct Option is A

Step 1:
In this case, we need to calculate the eigenvalues of \( A^{13} \). First, let's find the eigenvalues of matrix \( A \) by solving the characteristic equation.
- The eigenvalues of \( A \) are \( \lambda_1 = 3 \), \( \lambda_2 = -1 \), \( \lambda_3 = -1 \).

Step 2: Find the eigenvalues of \( A^{13} \).
- The eigenvalues of \( A^{13} \) are the eigenvalues of \( A \) raised to the power 13. Therefore, the eigenvalues of \( A^{13} \) are: \[ 3^{13}, \quad (-1)^{13} = -1, \quad (-1)^{13} = -1. \]

Thus, the correct answer is (A).