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Idempotent Matrix

  1. Nov 16, 2008 #1
    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. jcsd
  3. Nov 16, 2008 #2


    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:
    \begin{array} {c c}
    1 & 1 \\
    0 & 0
    \end{array} \right]
  4. Nov 16, 2008 #3
    THANK YOU DUDE !! u really helped me :)
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