Similarity of $2 \times 2$ matrices

Question

If i am not mistaken,

two $3 \times 3$ matrices are similar $ \iff \ $ they have the same characteristic and minimal polynomial.

Also, if two $3 \times 3$ matrices have the same characteristic polynomial, then it DOESN'T mean they are similar. Because they must agree also in minimal polynomial.

But how about in $2 \times 2$ matrices? Is it sufficient to say that two $2 \times 2$ matrices are similar if they have the same characteristic polynomial? Or they must agree also in minimal polynomial? If yes, can anyone provide a proof?

Thank you so much

Answer 1

The matrices $\left[\begin{smallmatrix}0&0\\0&0\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right]$ are not similar, in spite of the fact that they have the same characteristic polynomials.