1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Matrix proof

  1. Apr 24, 2009 #1
    1. The problem statement, all variables and given/known data
    Given a matrix B, if B = B2, is (B+I) invertible?

    2. The attempt at a solution

    det(B) = 0 or 1

    rref(rref(B) + I) is I, so (rref(B) + I) is invertible

    if det(B) = 1:
    let E1E2...En = B
    then E1E2...En(rref(B) + I) = B + E1E2...En

    I'm not sure if what I did is even useful =(
  2. jcsd
  3. Apr 24, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper

    What can you say about det(B + I)?
  4. Apr 25, 2009 #3
    If det(B) = 1, then B-1B = B-1B2, B = I
    so det(B+I) = det(2I) != 0, so B+I is invertible.

    I'm still stuck on if det(B) = 0..

    I'm quite sure that if B = B2, then B must be a diagonal matrix with entries being either 1 or 0, but I don't know how to prove it.
  5. Apr 26, 2009 #4


    User Avatar
    Science Advisor

    If B= B2 then B2- B= B(B- I)= 0.
  6. Apr 26, 2009 #5

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    That follows from the fact you know its minimal poly divides X^2-X, hence you know all the possible eigenvalues (0 and 1), asndyou know the geometric multiplicity is 0 or 1.

    Alternatively remember that a definition of an eigenvalue is:

    t is an eigenvalue of X if and only if X-tI is not invertible.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook