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QUESTION

If $A$ and $B$ are $n\times n$ matrices show that
tr$(AB)$=tr$(BA)$. Deduce that if two matrices are similar then
their traces are equal.

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ANSWER

tr$(AB)=\sum_r(AB)_{rr}=\sum_r\sum_s(A)_{rs}(B)_{sr}
=\sum_s\sum_r(B)_{sr}(A)_{rs}=\sum_s(BA)_{ss} $= tr$(BA)$

Hence
tr$(A^{-1}BA)=$tr$(A^{-1}(BA))$=tr$((BA)A^{-1})=$tr$(BAA^{-1})$=tr$(B)$.



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