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\begin{align*}

\sigma (P) &=\{ {\lambda \in C:\det (P(\lambda )) = 0} \} = \{ {\lambda \in C:\det (I\lambda - A) = 0}\} \\

&= \{ {\lambda \in C:0 \in \sigma (I\lambda - A)}\} =\{ {\lambda \in C:0 \in \lambda - \sigma (A)} \} = \sigma (A) 

\end{align*}





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