Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 7B / a

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Solution

The matrix trace is

Tr (A) = \sum_{i = 1}^{n} A_{ii} .

Using matrix multiplication and interchanging dummy indices,

Tr (A B) = \sum_{i, j} A_{ij} B_{ji} = \sum_{j, i} B_{ji} A_{ij} = Tr (B A) .

The proposed noncyclic interchange is false. With matrix units,

A = E_{11}, B = E_{12}, C = E_{21},

one has

Tr (A BC) = 1, Tr (A CB) = 0.

Trace is invariant under cyclic permutations, but an arbitrary swap need not be cyclic.

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