Critical points of matrix

[TanWu]

Homework Statement

Give their critical points of this system which can be written as a matrix:
$x^{\prime}=x - 2y, y^{\prime}=-2x+ 4y$

Relevant Equations

$x^{\prime}=x - 2y, y^{\prime}=-2x+ 4y$

My attempt is:

Condition for critical point is $x' = y' = 0$,
$0 = x - 2y \implies 2y = x$
$-2x + dy = 0$
Then $-4y + 4y = 0$

However, this means that critical points are $(2y, y)$ as system is linearly dependent (both equations are the same) where $y \in \mathbb{R}$. However, that means there are infinitely many critical points which I have a doubt about.

I express gratitude to those who help.

 

[docnet]

However, that means there are infinitely many critical points which I have a doubt about.

Why so?

 

[FactChecker]

Homework Statement: Give their critical points of this system which can be written as a matrix:
$x^{\prime}=x - 2y, y^{\prime}=-2x+ 4y$
Relevant Equations: $x^{\prime}=x - 2y, y^{\prime}=-2x+ 4y$

My attempt is:

Condition for critical point is $x' = y' = 0$,
$0 = x - 2y \implies 2y = x$
$-2x + dy = 0$

Do you mean ##-2x + 4y = 0## here?

Then $-4y + 4y = 0$

However, this means that critical points are $(2y, y)$ as system is linearly dependent (both equations are the same) where $y \in \mathbb{R}$. However, that means there are infinitely many critical points which I have a doubt about.

Your work looks ok to me. If you have doubts, have you checked that you got the initial problem equations right?

 

[TanWu]

Why so?

Do you mean ##-2x + 4y = 0## here?

Your work looks ok to me. If you have doubts, have you checked that you got the initial problem equations right?

Thank you Sirs. I apoglize, that is a typo of me. Yes, got the initial problem equations correct. I was only expecting one critical point.

 

[FactChecker]

Thank you Sirs. I apoglize, that is a typo of me.

Mathematics is very unforgiving in many ways. It's a learned skill to review your work very carefully.

Yes, got the initial problem equations correct. I was only expecting one critical point.

You did a good job! The problems where you get a different answer than you expected are ones that really test you.