Let's try a random-ish example.
$$ A = \pmatrix{0 & 0 & 1\cr 1 & -1 & -3\cr 1 & -2 & -1\cr}\ \text{over}\ \mathbb C$$
Its rank is $3$. The most obvious way to make the rank $2$ is by changing the top right $1$ to $0$, but you could change any of the entries $a_{ij}$ except $a_{23}$ and $a_{33}$ to make the rank $2$. On the other hand, to make the rank $0$ will require changing all seven nonzero entries to $0$. To make the rank $1$ will require changing at least $3$ entries, I think. For example, you could change $a_{13}$ to $0$, $a_{23}$ to $-1$ and $a_{32}$ to $-1$. So $\text{Rig}(A,3) = 0$, $\text{Rig}(A, 2) = 1$, $\text{Rig}(A,1) = 3$, $\text{Rig}(A,0) = 7$.