Equal traces and similar matrix

Question

I know that if matrix $a$ is similar to matrix $b$ then $\operatorname{trace} a=\operatorname{trace} b$.

Does it go to the other side?

Thanks.

Answer 1

No. The matrix $\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)$ and the null matrix are not similar.

Answer 2

The matrix $\left(\begin{smallmatrix}0&0\\0&0\end{smallmatrix}\right)$ and and the matrix $\left(\begin{smallmatrix}-1&0\\0&1\end{smallmatrix}\right)$ both have trace $0$, yet are clearly not similar. Thus the answer is no.