Counterexample
Let A=⎣⎡100020002⎦⎤. The matrix A is diagonal, hence diagonisable.
Find the eigenvalues of A
A−λI=⎣⎡1−λ0002−λ0002−λ⎦⎤
det(A−λI)=∣∣1−λ0002−λ0002−λ∣∣
=(1−λ)(2−λ)2=0 λ1=1,λ2=λ3=2.
The matrix A has eigenvalues 1,2,2 (not all distinct), but is diagonisable.
False.
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