Question

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.