Question

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

Answer 1

$MatrixAisanidempotentmatrixif{A}^2=A\left[\begin{array}{cc}a& b\\ c& 1-a\end{array}\right]\left[\begin{array}{cc}a& b\\ c& 1-a\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& 1-a\end{array}\right]a^2+bc=a\Rightarrow a^2+\frac{1}{4}=a\Rightarrow 4a^2-4a+1=0\Rightarrow a=\frac{1}{2}f\left(x\right)=x-x^2\Rightarrow f\left(a\right)=\frac{1}{4}Hence\frac{1}{f\left(a\right)}=4$