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NonCommAlg let $\displaystyle A=[a_{ij}]$ and suppose that $\displaystyle \{e_{ij} : \ 1 \leq i,j \leq n \}$ is the standard basis for the vector space of all $\displaystyle n \times n$ matrices over $\displaystyle F.$ now $\displaystyle \text{tr}(AB)=0,$ for all $\displaystyle B,$ if and only if $\displaystyle \text{tr}(Ae_{ij})=0,$ for all
$\displaystyle 1 \leq i,j \leq n.$ but $\displaystyle Ae_{ij}=\sum_{k=1}^n a_{ki}e_{kj}$ and thus $\displaystyle \text{tr}(Ae_{ij})=a_{ji}.$ therefore $\displaystyle \text{tr}(Ae_{ij})=0,$ for all $\displaystyle 1 \leq i,j \leq n,$ if and only if $\displaystyle A=0.$