The question: Diagonalisability of 2×2 matrices with repeated eigenvalues suggests that if a matrix has all its eigen values distinct, it must be diagonalizable. However, any multiple of the identity matrix will have all of its eigen values the same and yet be diagonalizable. I conjecture that if a general $n \times n$ matrix has some non-zero off diagonal elements, and has any multiplicity of eigen values, it will not be diagonalizable. I haven't been able to find a counterexample to this. Can it be proven (or disproven with a counter example).
EDIT: Sorry, I was actually looking for a stochastic matrix (rows must sum to one) with these properties. I'll add another - matrix has to be full rank (so no zero eigen values). If no one answers with such an example in the next few hours, I'll accept the current one.