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Is this true?

AB and BA have the same minimal polynomial

No — it is false.

The counterexample

A = [[0,1],[0,0]], B = [[0,0],[0,1]]

AB = A has minimal polynomial , BA = 0 has x. The characteristic polynomials do agree.

The kind of mistake this is

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Drill statements like this
linear algebra

Others that fail the same way

From Vector Spaces and Linear Maps › Linear transformations, matrix representation, change of basis

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