## Idempotent matrices

### Problem 572

A matrix $M$ is called idempotent if $M^2 = M$.

Let $M$ be a three by three matrix :
$M=\begin{pmatrix}
a & b & c\\
d & e & f\\
g &h &i\\
\end{pmatrix}$.

Let C(`n`) be the number of idempotent three by three matrices $M$ with integer elements such that

$ -n \le a,b,c,d,e,f,g,h,i \le n$.

C(1)=164 and C(2)=848.

Find C(200).