# Construct a 2x2 nilpotent matrix

• Aleoa
In summary, the conversation is about finding the equations for a nilpotent matrix and determining the minimum number of equations needed for two variables. The equations that have been discussed are trace = 0, determinant = 0, and kernel equations. It is argued that the trace and determinant equations are sufficient to determine a nilpotent matrix, while the kernel equations are implied by these equations. However, it is pointed out that the trace equation was already present and the determinant equation is implied by the kernel equations. The conversation ends with the question of how many equations are needed for two variables.
Aleoa

Costruct a:

## The Attempt at a Solution

I found 3 equations but i miss another one :(

$M=\begin{bmatrix} a & b\\ c & d \end{bmatrix}$

(1) a+d = 0 from the definition of nilpotent matrix
(2) a+3b = 0 from kernel
(3) c +3d= 0 from kernel

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Aleoa said:

## Homework Statement

Costruct a:

View attachment 224200

## The Attempt at a Solution

I found 3 equations but i miss another one :(

$M=\begin{bmatrix} a & b\\ c & d \end{bmatrix}$

(1) a+d = 0 from the definition of nilpotent matrix
(2) a+3b = 0 from kernel
(3) c +3d= 0 from kernel
You're missing an equation that results from ##M^2 = 0##.

Mark44 said:
You're missing an equation that results from ##M^2 = 0##.

I know that the equations of the nilpotent matrix are:

a+d=0
and

However, the first it's already present, the second it's implicitely represented by the 2 kernel equations

Aleoa said:
I know that the equations of the nilpotent matrix are:

a+d=0
and
Yes, you already said that in post #1, but you're not using the fact that the matrix is nilpotent.
Aleoa said:
However, the first it's already present, the second it's implicitely represented by the 2 kernel equations
Using the fact that tr(M) = 0, you have a + d = 0.
So M could be written as ##M = \begin{bmatrix} a & b \\ c & -a \end{bmatrix}##. Then ##M^2## = what?

Mark44 said:
Yes, you already said that in post #1, but you're not using the fact that the matrix is nilpotent.

Using the fact that tr(M) = 0, you have a + d = 0.
So M could be written as ##M = \begin{bmatrix} a & b \\ c & -a \end{bmatrix}##. Then ##M^2## = what?

The fact that the matrix is nilpotent is totally determined by the 2 equations:
tr = 0
det = 0

Maybe I am wrong, but the book I am studying argues this.

So, i miss another equation...

Last edited:
Aleoa said:
The fact that the matrix is nilpotent is totally determined by the 2 equations:
tr = 0
det = 0

Maybe I am wrong, but the book I am studying argues this.

So, i miss another equation...

You have enough equations now.

PeroK said:
You have enough equations now.

Unfortunately no. The tr=0 equation was already present and the det is implied by the 2 kernel equations

Aleoa said:
Unfortunately no. The tr equation was already present and the other it's implied by the 2 kernel equations

How many equations do you need for two variables?

The question does not imply that there is only one possible answer.

## What is a 2x2 nilpotent matrix?

A 2x2 nilpotent matrix is a square matrix with 2 rows and 2 columns that has the property that when raised to a certain power, the resulting matrix becomes a zero matrix.

## How can I construct a 2x2 nilpotent matrix?

To construct a 2x2 nilpotent matrix, you can start with a matrix with zeros in all entries except for the top right corner. Then, you can choose any non-zero number to fill in the bottom left entry. This will result in a matrix that, when squared, becomes a zero matrix.

## What is the significance of a nilpotent matrix in mathematics?

Nilpotent matrices are important in linear algebra and other areas of mathematics because they have unique properties that make them useful for solving equations and studying systems of equations.

## Can a 2x2 nilpotent matrix have any other non-zero entries?

No, a 2x2 nilpotent matrix can only have non-zero entries in the top right and bottom left positions. This is because any other non-zero entry would result in the matrix not being nilpotent.

## How is a nilpotent matrix different from a diagonal matrix?

A nilpotent matrix is different from a diagonal matrix in that a diagonal matrix can have non-zero entries in any position along the main diagonal, while a nilpotent matrix can only have non-zero entries in the top right and bottom left positions. Additionally, a diagonal matrix has distinct eigenvalues, while a nilpotent matrix has at least one eigenvalue of 0.

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