# Help with solution group of a Homogeneous system

• afik
In summary: But I don't know that, I just computed it.Do you know how to calculate it then?Yes, I can calculate it.In summary, the solution group of the system A^3X=0 is not equal to the solution group of the system AX=0. If this is true you will prove it, if not give a counterexample.
afik
Summary:: need help with solution group of Homogeneous system

Is the solution group of the system A^3X = 0
, Is equal to the solution group of the system AX = 0

If this is true you will prove it, if not give a counterexample.

thank you.

Have you tried anything?

I tried but I failed..
The matrix A ∈ Mn (R)

Well, you can show us your work, it could be a good point to start

second attempt:

Doesn't the first attempt exactly answer your question?

The second attempt has the problem that ##A^{-1}## doesn't exist for ##A \in M_n(\mathbb{R})##.

Gaussian97 said:
Doesn't the first attempt exactly answer your question?

The second attempt has the problem that ##A^{-1}## doesn't exist for ##A \in M_n(\mathbb{R})##.
I do not know if I do it right in the first attempt...
Why in the second attempt A^-1 does not exist?

Well, literally, your question is:
"Is the solution group of the system ##A^3 x = 0## equal to the solution group of the system ##Ax = 0##?"

And you literally wrote:
"system group of ##A x = 0 ## are not equal to system group of ##A^3 x = 0##"

For the ##A^{-1}## question is easy, the example you use, let ##A = \begin{pmatrix}0&1\\0&0\end{pmatrix}## then
$$A \in M_2(\mathbb{R}), \qquad \nexists A^{-1}$$

Gaussian97 said:
Well, literally, your question is:
"Is the solution group of the system ##A^3 x = 0## equal to the solution group of the system ##Ax = 0##?"

And you literally wrote:
"system group of ##A x = 0 ## are not equal to system group of ##A^3 x = 0##"

For the ##A^{-1}## question is easy, the example you use, let ##A = \begin{pmatrix}0&1\\0&0\end{pmatrix}## then
$$A \in M_2(\mathbb{R}), \qquad \nexists A^{-1}$$
I wrote:
"system group of ##A x = 0 ## are not equal to system group of ##A^3 x = 0##"

I wrote it down, but I do not know if my counter-argument is true there, or if I am wrong.

Ah well, then that's a completely different thing. What in that argument do you think can be wrong?

Gaussian97 said:
Ah well, then that's a completely different thing. What in that argument do you think can be wrong?
If I know what can be wrong I finish the homework... I don't know :(

Ok, so what things in your argument are you absolutely sure are correct?

If Ax= 0 then certainly A^3x=A^2(Ax)= A^2(0)= 0 for A any linear operator. So the solution set of A is a subset of the solution set of A^3. The question is "are there x that satisfy A^3x= 0 but not Ax= 0?

Can you help me more, I need it for tomorrow :(

Yes, that's correct. But in your prove you essentially did 3 things:
1. Calculate the solutions of ##Ax=0##
2. Calculate the matrix ##A^3##
3. Calculate the solutions of ##A^3 x = 0##

What of these steps do you think may be wrong? And what of these steps you are 100% sure are right?

I think that 1 and 3 are correct and 2 is wrong.

Well, so let's take a closer look at the computation of ##A^3##, do you have the intermediate steps?

Gaussian97 said:
Well, so let's take a closer look at the computation of ##A^3##, do you have the intermediate steps?
No, I don't have any idea.

Have you studied how matrices are multiplied? If you know that to compute ##A^3## is just computing first
$$A^2 = A \cdot A$$
and then
$$A^3 = A \cdot A^2$$

## 1. What is a homogeneous system?

A homogeneous system is a system of equations where all the terms have the same degree. This means that all the variables in the equations have the same exponent.

## 2. How do you solve a homogeneous system?

To solve a homogeneous system, you can use the method of substitution or elimination. You can also use matrices and Gaussian elimination to find the solution.

## 3. What is the solution group of a homogeneous system?

The solution group of a homogeneous system is the set of all possible solutions to the system of equations. It can be represented as a vector space or a set of linear combinations of the equations.

## 4. Can a homogeneous system have more than one solution?

Yes, a homogeneous system can have infinitely many solutions. This is because the equations in a homogeneous system are linearly dependent, meaning that any scalar multiple of a solution is also a solution.

## 5. How do you know if a homogeneous system has no solution?

A homogeneous system has no solution if the equations are inconsistent, meaning that there is no set of values for the variables that satisfy all the equations. This can be determined by using Gaussian elimination to reduce the system to row echelon form.

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