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**JJMC89** a) Show that if $\displaystyle A$ and $\displaystyle B$ are $\displaystyle n$-by-$\displaystyle n$ matrices with coefficients in an arbitrary field $\displaystyle F$, then $\displaystyle Tr(AB) = Tr(BA)$.

b) use (a) to show that if $\displaystyle A$ and $\displaystyle B$ are similar matrices, then $\displaystyle Tr(A) = Tr(B)$.

c) Show that the matrices of trace zero form a subspace of $\displaystyle M{_n}(F)$ of dimension $\displaystyle n{^2} - 1$. [Hint: The mapping $\displaystyle Tr: M{_n}(F) \rightarrow F$ is a linear transformation.]

d) Prove that the subspace of $\displaystyle M{_n}(F)$ defined in (c) is generated by the matrices $\displaystyle AB - BA,\ A,\ B \in M{_n}(F)$.