Upgrade to access all content for this subject

Let $A$ be an $n \times n$ matrix and let $k$ be a positive integer.

What is the relationship between the eigenvalues of $A$ and the eigenvalues of $A^k$?

If $\lambda$ is an eigenvalue of $A$, then $\lambda$ is an eigenvalue of $A^k$.

If $\lambda$ is an eigenvalue of $A$, then $\lambda^k$ is an eigenvalue of $A^k$.

If $\lambda$ is an eigenvalue of $A$, then $k \lambda$ is an eigenvalue of $A^k$.

If $\lambda$ is an eigenvalue of $A$, then $\lambda^{1/k}$ is an eigenvalue of $A^k$.

None of the above.