Determine if Matrix is Triangularizable or Diagonalizable

Example

With $A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$ Then, $det(A)=x^2+1$ As the discriminant is negative, $x^2+1$ is a prime factor, Then, $f(x)$ is not a product of prime factors Therefore, $T$ is not Triangularizable or Diagonalizable over $\mathbb{R}$.

Example 2