# 3.II.10H

(a) Define what is meant by the trace of a complex $n \times n$ matrix $A$. If $T$ denotes an $n \times n$ invertible matrix, show that $A$ and $T A T^{-1}$ have the same trace.

(b) If $\lambda_{1}, \ldots, \lambda_{r}$ are distinct non-zero complex numbers, show that the endomorphism of $\mathbf{C}^{r}$ defined by the matrix

$\Lambda=\left(\begin{array}{ccc} \lambda_{1} & \ldots & \lambda_{1}^{r} \\ \vdots & \ldots & \vdots \\ \lambda_{r} & \ldots & \lambda_{r}^{r} \end{array}\right)$

has trivial kernel, and hence that the same is true for the transposed matrix $\Lambda^{t}$.

For arbitrary complex numbers $\lambda_{1}, \ldots, \lambda_{n}$, show that the vector $(1, \ldots, 1)^{t}$ is not in the kernel of the endomorphism of $\mathbf{C}^{n}$ defined by the matrix

$\left(\begin{array}{ccc} \lambda_{1} & \ldots & \lambda_{n} \\ \vdots & \ldots & \vdots \\ \lambda_{1}^{n} & \ldots & \lambda_{n}^{n} \end{array}\right)$

unless all the $\lambda_{i}$ are zero.

[Hint: reduce to the case when $\lambda_{1}, \ldots, \lambda_{r}$ are distinct non-zero complex numbers, with $r \leqslant n$, and each $\lambda_{j}$ for $j>r$ is either zero or equal to some $\lambda_{i}$ with $i \leqslant r$. If the kernel of the endomorphism contains $(1, \ldots, 1)^{t}$, show that it also contains a vector of the form $\left(m_{1}, \ldots, m_{r}, 0, \ldots, 0\right)^{t}$ with the $m_{i}$ strictly positive integers.]

(c) Assuming the fact that any complex $n \times n$ matrix is conjugate to an uppertriangular one, prove that if $A$ is an $n \times n$ matrix such that $A^{k}$ has zero trace for all $1 \leqslant k \leqslant n$, then $A^{n}=0 .$