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: Nilpotent Matrix Proof

  1. Mar 14, 2012 #1
    1. The problem statement, all variables and given/known data
    A. Are nilpotent matrices invertible ? Prove your answer.

    B. If A is nilpotent, what can you say about (A)^τ ? Prove your answer.

    C. If A is nilpotent, show I-A is invertible.

    2. Relevant equations


    3. The attempt at a solution

    A. I know invertible matrix are - AB = BA = I

    B. I took a nilpotent matrix
    A = [ 0 1
    0 0 ]
    Its transpose is -
    (A)^τ = [ 0 0
    1 0 ]
    And the transpose is still a nilpotent matrix because
    (A^τ)^2 = [ 0 0
    0 0 ]
    But I dont know if its true for all and it says prove your answer.

    C. No idea
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Mar 14, 2012 #2
    A matrix is invertible if and only if it has non-zero determinant. What can you say about the determinant of a nilpotent matrix?
    C. You could prove this by assuming it is false i.e. I-A is not invertible and then proceeding.
  4. Mar 14, 2012 #3
    A nilpotent matrix has determinant 0 since its diagonals are all 0 (Eigen values are 0). So the inverse would be 0 too .
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook