[Math] Are two matrices of the same rank similar

Question

I know that if two matrices $A$ and $B$ are similar implies that they have the same rank.

However, if they have the same rank are they similar?

Answer 1

As mentioned by the others, the answer is negative. Actually we can say something more: if $n\ge2$, then for any $n\times n$ nonzero matrix $A$, there is always a dissimilar matrix $B$ of the same rank; if $n=1$, the statement also holds when the characteristic of the field is not $2$.

Proof. The case $n=1$ is trivial. Suppose $n\ge2$. Let $k=\operatorname{rank}(A)$. If $A$ is not diagonalisable, let $B$ be a diagonal matrix of rank $k$. If $A$ is diagonalisable, let $B$ be the direct sum of a $k\times k$ Jordan block for eigenvalue $1$ and a zero block.