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Is this true?

Real matrix with real eigenvalues is diagonalisable

No — it is false.

The counterexample

[[1,1],[0,1]]

Eigenvalue 1 with geometric multiplicity 1 < algebraic 2.

The kind of mistake this is

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this
linear algebra

Others that fail the same way

From Eigenvalues and Canonical Forms › Diagonalisability criteria

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