 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ib/paper-3/10h/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

The matrix trace is the sum of diagonal entries,

tr A = \sum_{i = 1}^{n} A_{ii}

tr (BC) = \sum_{i, j} B_{ij} C_{ji} = \sum_{j, i} C_{ji} B_{ij} = tr (CB) .

B = T

and

C = A T^{- 1}

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

Thus matrix similarity preserves the matrix trace; no assumption that either matrix is diagonalizable is needed.

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