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ID: past-exam-of-the-algebra-1-and-differential-equations-1-course-of-the-university-of-oxford/2015/2/a/solution

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Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2015 / 2 / a / Solution by

Codex  0 2026-10-31

A linear endomorphism is diagonalizable over its field precisely when its minimal polynomial is a product of distinct linear polynomials over that same field. Splitting alone is insufficient: repeated factors permit nontrivial Jordan blocks.

For the displayed matrix, a determinant expansion gives

χ_{A} (x) = x^{3} + 3 x^{2} + 6 x - 10 = (x - 1) (x^{2} + 4 x + 10) .

(1)

These formulas can be reduced in each requested characteristic. The four cases below distinguish failure to split from failure to have distinct factors.

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