- #1

brownman

- 13

- 0

## Homework Statement

If A

^{2}is a zero matrix, find all symmetric 2x2 nilpotent matrices.

## Homework Equations

## The Attempt at a Solution

So if A

^{2}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, a

^{2}must equal negative b

^{2}, so there is no solution. Is there no 2x2 symmetric nilpotent matrices, or did I mess up somewhere?