If $\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& 1-\mathrm{a}\end{array}\right]$ is an idempotent matrix and f(x) = x - x² and bc = 1/4, then the value of f(a) is __________.

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

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

$\begin{array}{l}\mathrm{or} \left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& 1-\mathrm{a}\end{array}\right]\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& 1-\mathrm{a}\end{array}\right]=\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& 1-\mathrm{a}\end{array}\right]\\ \mathrm{or} \left[\begin{array}{cc}{\mathrm{a}}^2+\mathrm{bc}& \mathrm{ab}+\mathrm{b}-\mathrm{ab}\\ \mathrm{ac}+\mathrm{c}-\mathrm{ac}& \mathrm{bc}+(1-\mathrm{a}{)}^2\end{array}\right]=\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& 1-\mathrm{a}\end{array}\right]\\ \mathrm{or} \left[\begin{array}{cc}{\mathrm{a}}^2+\mathrm{bc}& \mathrm{b}\\ \mathrm{c}& \mathrm{bc}+(1-\mathrm{a}{)}^2\end{array}\right]=\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& 1-\mathrm{a}\end{array}\right]\\ \mathrm{or} {\mathrm{a}}^2+\mathrm{bc}=\mathrm{a}\\ \mathrm{a}-{\mathrm{a}}^2=\mathrm{bc}=1/4 \left(\mathrm{given}\right)\\ \mathrm{f}\left(\mathrm{a}\right)=1/4\end{array}$