Help with solution group of a Homogeneous system

[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.

 

[Gaussian97]

Have you tried anything?

 

[afik]

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

 

[Gaussian97]

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

 

[afik]

second attempt:

 

[Gaussian97]

Doesn't the first attempt exactly answer your question?

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

 

[afik]

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?

 

[Gaussian97]

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 $A^3 x = 0$"

So doesn't that answer your question?

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

 

[afik]

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 $A^3 x = 0$"

So doesn't that answer your question?

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.

 

[Gaussian97]

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

 

[afik]

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 :(

 

[Gaussian97]

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

 

[afik]

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 :(

 

[Gaussian97]

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?

 

[afik]

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

 

[Gaussian97]

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

 

[afik]

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.

 

[Gaussian97]

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
$$A^3 = A \cdot A^2$$