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[SPECTRAL MAPPING THEOREM] Let $A$ be a square matrix.
a) If $A(A-I)(A-2 I)=0$, show that the only possible eigenvalues of $A$ are $\lambda=0$, $\lambda=1$, and $\lambda=2$.
b) Let $p$ any polynomial. Show that the eigenvalues of the matrix $p(A)$ are precisely the numbers $p\left(\lambda_{j}\right)$, where the $\lambda_{j}$ are the eigenvalues of $A$.
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