Dimension of a specific subspace of $M_n(\mathbb{R})$

Question

If $A$ is an $n\times n$ matrix, then what is the dimension of all matrices $B$ of order $n\times n$ for which $AB = 0$ holds.

I know that the column space of such $B$ must be contained in $Nul(A).$ After that unable to think any way out. Any help. Thank you

Answer 1

It is $n \cdot \dim(\text{Nul}(A))$. One way to see this is to view $B$ as a $n^2 \times 1$ vector by stacking its columns, and note that you are asking for this large vector to be in the nullspace of the block diagonal $n^2 \times n^2$ matrix with copies of $A$ along its [block] diagonal.