Question

The matrix \( A = \begin{bmatrix} 3 & -1 & 1 \\ 1 & -5 & 1 \\ 1 & -1 & 3 \end{bmatrix} \) has eigenvalues 2, 3, 6 then the eigenvalues of \( A^4 \) are

• A. \( 2, \frac{4}{3}, \frac{2}{3} \)
• B. \( 2, 3, 6 \)
• C. \( 8, 12, 24 \)
• D. \( \frac{1}{2}, \frac{1}{3}, \frac{1}{6} \)

Hint:

To find the eigenvalues of \( A^n \), raise the eigenvalues of \( A \) to the power \( n \).

Answer 1

The Correct Option is A

The eigenvalues of \( A^n \) are simply the eigenvalues of \( A \) raised to the power \( n \). For \( A^4 \), the eigenvalues are \( 2^4, 3^4, 6^4 \). Therefore, the eigenvalues of \( A^4 \) are \( 2, \frac{4}{3}, \frac{2}{3} \).