Discussion
Which of the following statements is true ?
Answer: A
The true statement is Trace (AB) = Trace (BA).
\[ \begin{aligned} \mathrm{Tr}(AB) &= \sum_{i}\sum_{j} A_{ij}B_{ji} \\ &= \sum_{j}\sum_{i} B_{ji}A_{ij} = \mathrm{Tr}(BA) \end{aligned} \]
More generally the trace is unchanged by cyclic permutations: Tr(ABC) = Tr(BCA) = Tr(CAB). Swapping two neighbouring factors in a longer product, as in Tr(BAC) or Tr(BACD), is not a cyclic permutation, so those equalities fail for general matrices. Tr(AB) = Tr(A) Tr(B) is false too: for A = B = the 2 × 2 identity, Tr(AB) = 2 while Tr(A) Tr(B) = 4.
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