If $\left[\begin{array}{cc}a& b\\ c& 1-a\end{array}\right]$  is an idempotent matrix and $f\left(x\right)=x-x^2$ and  $bc=\frac{1}{4}$ then the value of $\frac{1}{f\left(a\right)}$ is ------

$\left[\begin{array}{cc}a& b\\ c& 1-a\end{array}\right]\text{ is an idempotent matrix. }$

$\Rightarrow {\left[\begin{array}{cc}a& b\\ c& 1-a\end{array}\right]}^2=\left[\begin{array}{cc}a& b\\ c& 1-a\end{array}\right]$

$\text{ or }\left[\begin{array}{cc}a& b\\ c& 1-a\end{array}\right]\left[\begin{array}{cc}a& b\\ c& 1-a\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& 1-a\end{array}\right]$

$\text{ ог }\left[\begin{array}{cc}{a}^2+bc& b\\ c& bc+(1-a{)}^2\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& 1-a\end{array}\right]$

$\text{ or }{a}^2+bc=a\Rightarrow a-a^2=bc=1/4$

$f\left(a\right)=1/4$