[Math] Example of 2 matrices similar but not row equivalent

Question

If two matrices are row equivalent, they may not be similar because all invertible matrices are row equivalent to $I$, yet not all invertible matrices have the same trace, eigenvalues etc.

Is it also true that if two matrices are similar, they may not be row equivalent? My instinct is that there is no reason that 2 similar matrices need to be row equivalent since having the same rank, eigenvalues, determinant etc does not necessarily make them row equivalent.

Any suggestions as to how to find a counter-example?

Thanks for help.

Answer 1

$$\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}\begin{bmatrix} 0 & 1 \\ 0 & 0\end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}=\begin{bmatrix} 0 & 0 \\ 1 & 0\end{bmatrix}$$

$\begin{bmatrix} 0 & 1 \\ 0 & 0\end{bmatrix}$ and $\begin{bmatrix} 0 & 0 \\ 1 & 0\end{bmatrix}$ are not row equivalent though they are similar.