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 \( A^{13} \).
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 \( A^{13} \).
Show Hint
Eigenvalues of \( A^{13} \) are powers of the eigenvalues of \( A \).
Eigenvalues of \( A^{13} \) are the eigenvalues of \( A \) raised to the power of 13.
Eigenvalues of \( A^{13} \) are the eigenvalues of \( A \) raised to the power of 13.
Eigenvalues of \( A^{13} \) are the eigenvalues of \( A \) raised to the power of 13.
The Correct Option is A
Solution and Explanation
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).
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