Does similar matrices have same column space or row space

Question

I am reading similar matrices and I found they share many properties like char. Polynomial, trace, eigenvalues, etc. I am curious to know whether they also have same column space or row space?

My intuition is NO.

"I know two equivalent matrices have same row space but may not have same column space(using elementary matrices)

But how to approach the above problem I have no idea.

Thanks in advance.

Answer 1

A= $\begin{pmatrix} 1&0\\ 0&0 \end{pmatrix}$ and B= $\begin{pmatrix} 0&0\\ 0&1 \end{pmatrix}$ both are diagonalisable Matrix and similar too. But both have different columns space and row space.