# Finding all 2x2 nilpotent matrices

1. Jan 27, 2013

### brownman

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. Jan 27, 2013

### 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.

3. Jan 27, 2013

### brownman

Oh yeah there is the trivial solution. Thanks for the help in clearing it up :)