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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2001/paper-1/10/a/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 1 / 10 / a / Solution by

Codex  0 2026-10-20

Use the Einstein summation convention, with repeated component indices summed. The following three expressions are listed separately below. For cyclic invariance of the trace,

tr (A BC) = a_{ij} b_{jk} c_{ki} = c_{ki} a_{ij} b_{jk} = tr (C A B),

(1)

where the middle equality commutes scalar entries, not the matrices themselves. If

A

is invertible, this gives

tr (A B A^{- 1}) = tr (B A^{- 1} A) = tr B .

(2)

This is similarity invariance of trace.

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