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# Problem 572

## Idempotent matrices

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).

## 幂等矩阵

$M=\begin{pmatrix} a & b & c\ d & e & f\ g &h &i\ \end{pmatrix}$。

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