Finding all 2x2 nilpotent matrices

[brownman]

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?

 

[Mark44]

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

What they're saying is that A is nilpotent. A² is the 2 x 2 zero matrix.

, then

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

You're looking for symmetric 2 x 2 matrices, which means they have to look like this:

$$ \begin{bmatrix} a & b \\ b & c\end{bmatrix}$$

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.

There is a solution.

Is there no 2x2 symmetric nilpotent matrices, or did I mess up somewhere?

 

[brownman]

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