Finding all 2x2 nilpotent matrices

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In summary, the question asks to find all symmetric 2x2 nilpotent matrices given that A2 is a zero matrix. The attempt at a solution involves multiplying the matrices and setting each element equal to zero. The first equation shows that a2 must equal negative b2, but there is a trivial solution where a=b=0. Therefore, there is at least one 2x2 symmetric nilpotent matrix.
  • #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?
 
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  • #2
brownman said:

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.
brownman said:
, 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:

$$ \begin{bmatrix} a & b \\ b & c\end{bmatrix}$$
brownman said:
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.
brownman said:
Is there no 2x2 symmetric nilpotent matrices, or did I mess up somewhere?
 
  • #3
Oh yeah there is the trivial solution. Thanks for the help in clearing it up :)
 

FAQ: Finding all 2x2 nilpotent matrices

1. What is a nilpotent matrix?

A nilpotent matrix is a square matrix where there exists a positive integer k such that when the matrix is raised to the power of k, it becomes the zero matrix.

2. How can I determine if a 2x2 matrix is nilpotent?

A 2x2 matrix is nilpotent if its trace (the sum of its diagonal elements) is zero and its determinant is also zero.

3. How many 2x2 nilpotent matrices are there?

There are infinitely many 2x2 nilpotent matrices, as any matrix with a trace of zero and determinant of zero can be considered nilpotent.

4. Can a nilpotent matrix have all non-zero entries?

No, a nilpotent matrix must have at least one zero entry in order to satisfy the condition of being raised to the power of k to become the zero matrix.

5. What are some real-world applications of nilpotent matrices?

Nilpotent matrices are commonly used in linear algebra and differential equations to represent systems that reach a steady state or equilibrium. They can also be used in cryptography for data encryption and decryption.

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