A standard notation: By $\displaystyle M_n(R)$ we mean the set of all $\displaystyle n \times n$ matrices with entries from some set $\displaystyle R.$

Problem: Let $\displaystyle A \in M_n(\mathbb{C}).$ Suppose that $\displaystyle \text{tr}(A)=\text{tr}(A^2)=\cdots = \text{tr}(A^n)=0.$ Prove that $\displaystyle A^n=\bold{0}.$

Remark: The above result remains true if we replace $\displaystyle \mathbb{C}$ with any field $\displaystyle F$ of characteristic 0. It is not necessarily true if $\displaystyle \text{char} F \neq 0.$ A counter-example is: $\displaystyle A=\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \in M_2(\mathbb{F}_2).$