Diagonalizability and triangularizability

Question

Let $A \in M_3(\mathbb{R})$ and let $X= \{ C∈\operatorname{GL}_3(\mathbb{R})|CAC^{−1}\text{ is triangular}\}$. Then

$A)$ $X \ne \emptyset$

$B)$ If $X=\emptyset$, then A is not diagonalizable over $\mathbb{C}$.

$C)$ If $X=\emptyset$, then A is diagonalizable over $\mathbb{C}$.

$D)$ If $X=\emptyset$, then A has no real eigenvalues.

I know the concept of diagonalizability but I am unable to find the correct option in this question. Please provide me some hint/explanation on how to proceed in this question.

Answer 1

In fact, the answer to this question is C.

If $A$ only has real eigenvalues, then $A$ has a Jordan canonical form with real entries, which means that $X$ is non-empty. So, if $X = \emptyset$, then $A$ must have at least one non-real eigenvalue.

Now, using the fact that the non-real eigenvalues of a real matrix come in conjugate pairs, conclude that the (complex) eigenvalues of $A$ must be distinct. Thus, $A$ is diagonalizable over $\Bbb C$.