Number Of Solutions $X^{2}=X$ The equation $x^{2}=x$ $,x\in \mathbb{R}$ has only the solutions $x=0$ and $x=1$.
For a $2 \times 2$ matrix X, how many solutions does $X^{2}=X$ have?
What if $X$ is symmetric?

Let $\textbf{X} = \left( {\matrix{
   a & b  \cr 
   c & d  \cr 
 } } \right)$, then  ${\textbf{X}}^2-\textbf{X} =\textbf{X}(\textbf{X} - \textbf{I})= 0$ is equivalent to 
$$ \left( {\matrix{
   a & b  \cr 
   c & d  \cr 
 } } \right)\left( {\matrix{
   a-1 & b  \cr 
   c & d-1  \cr 
 } } \right) = 0$$
Solve this equation for $a,b,c,d$, I get that 
$\left\{\left\{c=\frac{a-a^2}{b},d=1-a\right\},\{a=0,b=0,d=1\},\{a=1,b=0,d=0\},\{a=0,b=0,c=0,d=0\},\{a=0,b=0,c=0,d=1\},\{a=1,b=0,c=0,d=0\},\{a=1,b=0,c=0,d=1\}\right\}$
Obviously, there are infinite solutions. 
