Nilpotent matrices over a ring

Question

Any suggestions or ideas what method can be used to count the number of $2 \times 2$ nilpotent matrices over a ring $\mathbb{Z}_p [i]$ which is a prime field.

Answer 1

The nonzero nilpotent $2\times 2$ matrices are the conjugates of $A=\pmatrix{0&1\\0&0}$. The matrices commuting with $A$ are those of the form $rI+sA$, and there are $q(q-1)$ of these which are invertible when we are working over the finite field of order $q$. So $A$ has $|\textrm{GL}_2(q)|/(q(q-1))=q^2-1$ conjugates, so overall there are $q^2$ nilpotent matrices.