Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 8F / b

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- Solution b

Solution

False. At eigenvalue

2

, algebraic multiplicity two and minimal-polynomial exponent two force one block

J_{2} (2)

. At eigenvalue

1

, the blocks have total size five and largest size two, so there are only two possible partitions,

2 + 2 + 1 or 2 + 1 + 1 + 1.

Hence there are exactly two conjugacy classes satisfying the given polynomial data, represented by

J_{2} (2) \oplus J_{2} (1) \oplus J_{2} (1) \oplus J_{1} (1)

and

J_{2} (2) \oplus J_{2} (1) \oplus J_{1} (1) \oplus J_{1} (1) \oplus J_{1} (1) .

There cannot be three mutually non-conjugate examples.

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