 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ia/paper-1/6c/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

B = S^{- 1} A S

, then

det (t I - B) = det (S^{- 1} (t I - A) S) = det (t I - A),

so the characteristic polynomials, eigenvalues, and algebraic multiplicities agree. The converse is false:

I_{2}

and

(1011)

have the same characteristic polynomial, but the first is diagonalisable and the second is not, so they cannot be similar.

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