If `[a b c1-a]` is an idempotent matrix and `f(x)=x-^2=b c=1//4` , then the value of `1//f(a)` is ______.

Answer

0.25

Explanation

`[(a,b),(c,1-a)]` is an idempotent matrix.
`implies [(a,b),(c,1-a)]^(2)=[(a,b),(c,1-a)]`
or `[(a,b),(c,1-a)][(a,b),(c,1-a)]=[(a,b),(c,1-a)]`
or `[(a^(2)+bc,ab+b-ab),(ac+c-ac,bc+(1-a)^(2))]=[(a,b),(c,1-a)]`
or `[(a^(2)+bc,b),(c,bc+(1-a)^(2))]=[(a,b),(c,1-a)]`
or `a^(2)+bc=a`
`a-a^(2)=bc=1//4` (given)
`f(a)=1//4`