- #1
brownman
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Homework Statement
If A2 is a zero matrix, find all symmetric 2x2 nilpotent matrices.
Homework Equations
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?