Let $$A = \pmatrix{7&9\\-3&-5},$$
it is a $2\times 2$ matrix. For every integer $m$, find all eigenvalues of $A^m$, and bases for the corresponding eigenspaces
How to get it?!!
2026-04-19 03:14:11.1776568451
find the eigenvalue of $A^m$
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Hint If $\lambda$ is an eigenvalue of $A$ associated to the eigenvector $v$, then $$A^mv = A^{m-1}Av = A^{m-1}\lambda v = \ldots = \lambda^m v,$$ i.e. $\lambda^m$ is an eigenvalue of $A^m$. Now, can you find the eigenvalues of $A$?