 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/17/a/solution

By the definition of matrix multiplication and trace, with all indices summed from

1

D

tr (PQR) = \sum_{i, j, k} P_{ij} Q_{jk} R_{ki} .

The entries are scalars, so their multiplication commutes. Reordering them and renaming the summed indices gives

\sum_{i, j, k} Q_{jk} R_{ki} P_{ij} = \sum_{j, k, i} Q_{jk} R_{ki} P_{ij} = tr (QRP) .

Equivalently, in Einstein notation,

P_{ij} Q_{jk} R_{ki} = Q_{jk} R_{ki} P_{ij}

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