Idempotent Matrix

1. Nov 16, 2008

JosephR

1. The problem statement, all variables and given/known data
A martix A is said to idempotent if A²=A prove the following:

a) If A is idempotent then I-A is also idempotent
b) If A is idempotent and invertible then A=I
c) If A is idempotent then I-2A is invertible.Find (I-2A)-1 in terms of A
d)Give an example of a 2x2 idempotent matrix A such that A is not the zero or the identity matrix

2. The attempt at a solution

a) A²=A then,(I-A)(I-A)=I-A ( prove it)
= I²-2A+A² and since A²=A then I-A=I-A
b) A²=A
A.A=A ~~> A-1.A.A=A-1.A ~>IA=I ~>A=I
c) I'm Stuck

d) i knew this part :P

so plz any help on part c) i have an exam tomorow :)

Last edited: Nov 16, 2008
2. Nov 16, 2008

Staff: Mentor

c) Just to see what happens, I multiplied I - 2A by itself, and got (I - 2A)(I - 2A) = I
d) Here's an idempotent 2 x 2 matrix:
$$\left[ \begin{array} {c c} 1 & 1 \\ 0 & 0 \end{array} \right]$$

3. Nov 16, 2008

JosephR

THANK YOU DUDE !! u really helped me :)