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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

    Mark44

    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 :)
     
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