How to calculate the Zariski closure of $Z(A)B$?

Question

Suppose $A$ is a $n\times n$ matrix, and $Z(A)$ is the centralizer of $A$ in $M_{n\times n}$. If $B\in M_{n\times m}$, then how to calculate the Zariski closure of $Z(A)B$ in $M_{n\times m}$?

Answer 1

Note that $Z(A)$ is a sub-vector space of $M_{n\times n}$, and that $X \mapsto XB$ is a linear transformation $M_{n\times n} \to M_{n\times m}$, so it sends subspaces to subspaces.

This should be enough for you to figure out the rest.

$Z(A)B$ is a subspace so it is already Zariski closed.