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: Finding all 2x2 nilpotent matrices

  1. Jan 27, 2013 #1
    1. The problem statement, all variables and given/known data

    If A2 is a zero matrix, find all symmetric 2x2 nilpotent matrices.

    2. Relevant equations

    3. The attempt at a solution

    So if A2 is nilpotent, then

    [a,b;c,d]*[a,b;c,d] is equal to [0,0;0,0].

    Since A is symmetric, b=c. Multiplying the two matrices, I get

    [ aa + bb, ab + bd; ba +db, bb + dd] = [0,0;0,0]

    each element in the matrix must equal zero, so

    aa + bb = 0
    ab + bd = 0
    ba + bd = 0
    bb + dd = 0

    with the first equation, a2 must equal negative b2, so there is no solution. Is there no 2x2 symmetric nilpotent matrices, or did I mess up somewhere?
  2. jcsd
  3. Jan 27, 2013 #2


    Staff: Mentor

    What they're saying is that A is nilpotent. A2 is the 2 x 2 zero matrix.
    You're looking for symmetric 2 x 2 matrices, which means they have to look like this:

    $$ \begin{bmatrix} a & b \\ b & c\end{bmatrix}$$
    There is a solution.
  4. Jan 27, 2013 #3
    Oh yeah there is the trivial solution. Thanks for the help in clearing it up :)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook