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An $n \times n$ matrix is called nilpotent if $A^{k}$ equals the zero matrix for some positive integer $k$. (For instance, $\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)$ is nilpotent.)
a) If $\lambda$ is an eigenvalue of a nilpotent matrix $A$, show that $\lambda=0$. (Hint: start with the equation $A \vec{x}=\lambda \vec{x}$.)
b) Show that if $A$ is both nilpotent and diagonalizable, then $A$ is the zero matrix. [Hint: use Part a).]
c) Let $A$ be the matrix that represents $T: \mathcal{P}_{5} \rightarrow \mathcal{P}_{5}$ (polynomials of degree at most 5) given by differentiation: $T p=d p / d x$. Without doing any computations, explain why $A$ must be nilpotent.
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