Under what assumptions is it correct to say "a matrix is diagonalizable if and only if its eigenvalues are real"?

Question

A $2\times 2$ matrix is diagonalizable if and only if its eigenvalues are real.

Which statement is most correct:

1. The proposition is true only if the eigenvalues are all greater than zero.
2. The proposition is false. You also need the matrix to be orthogonal.
3. The proposition is false. You also need the algebraic dimension equal to the geometric dimension.
4. The proposition is true for $2\times 2$ matrices but not in general.
5. The proposition is false. You also need the matrix to be symmetric.

I think 2 is the answer, based on what I know that a matrix $A$ is diagonalizable if it is similar to a diagonal matrix.

If you could answer and explain why I would be most grateful.

Answer 1

The proposition, as I understand it, is

A $2\times 2$ matrix is diagonalizable (over the reals) if and only if its eigenvalues are real.

Of the statements given, 3) is the most correct. That is, the proposition is false: a matrix with only real eigenvalues is only diagonalizable if all algebraic and geometric dimensions are equal.

Answer 2

Every symmtric matrix is always diagonalizable and eigenvalues of symmtric matrix are always real so 5 is most correct.