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- Jan 26, 2012

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**Problem**: Let $V$ be a finite dimensional complex vector space. Let $\phi$ be an element of $\text{End}_{\mathbb{C}}(V)$, and consider the function $f:\mathbb{C}\rightarrow\mathbb{C}$ by \[f(z)=\det(1+z\cdot\phi).\]

Find an expression for $f^{\prime}(0)$ using what is known about trace, determinants, and the characteristic polynomial.

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