Construct a 2x2 nilpotent matrix

[Aleoa]

Homework Statement

Costruct a:

The Attempt at a Solution

I found 3 equations but i miss another one :(

<br /> M=\begin{bmatrix}<br /> a &amp; b\\<br /> c &amp; d<br /> \end{bmatrix}

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

 

[Mark44]

Homework Statement

Costruct a:

View attachment 224200

The Attempt at a Solution

I found 3 equations but i miss another one :(

<br /> M=\begin{bmatrix}<br /> a &amp; b\\<br /> c &amp; d<br /> \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 $M^2 = 0$.

 

[Aleoa]

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
ad=bc.

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

 

[Mark44]

I know that the equations of the nilpotent matrix are:

a+d=0
and
ad=bc.

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

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?

 

[Aleoa]

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...

 

[PeroK]

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.

 

[Aleoa]

You have enough equations now.

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

 

[PeroK]

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?

 

[TSny]

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