Inconsistent Linear Systems: Solving for k

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Homework Help Overview

The discussion revolves around determining the value(s) of k for which a linear system represented by an augmented matrix is inconsistent, meaning it has no solutions. Participants are exploring the implications of the system's equations and the conditions under which inconsistency arises.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the relationship between the determinant of the matrix and inconsistency, with some suggesting that a zero determinant may indicate inconsistency. Others question the original poster's understanding of the system and its representation.

Discussion Status

There is an ongoing exploration of the conditions for inconsistency, with various interpretations of the equations being examined. Some participants have provided guidance on how to approach the problem, while others have raised questions about the setup and assumptions made by the original poster.

Contextual Notes

Participants note the need for clarity regarding the system of equations represented by the augmented matrix, as well as the importance of understanding the implications of the determinant in this context.

Math100
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Homework Statement


For what value(s) of k is the linear system with augmented matrix inconsistent?

Homework Equations


None.

The Attempt at a Solution


I know that inconsistent means no solution. So do I set k=2k and solve for k?
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The most direct way to declare that a system is inconsistent is by looking for the value of ##k## which causes the determinant of the matrix to be zero. Since this is a fairly simple system however, you could easily pick a value of ##k## which causes one of the rows to be a multiple of the other.
 
Math100 said:

Homework Statement


For what value(s) of k is the linear system with augmented matrix inconsistent?

Homework Equations


None.

The Attempt at a Solution


I know that inconsistent means no solution. So do I set k=2k and solve for k?View attachment 219506
No.
What is the system of equations that your augmented matrix represents?

Also, what does the thing that looks like a backwards S?

Edit: Your attempt at a solution is just barely enough to get by. Other mentors would not have been as generous. In future posts, please make more of an attempt at solving the problems you're asking about.
 
k+2=1
1+2k=1
 
Math100 said:
k+2=1
1+2k=1
So what value of k would make it so that this system of equations doesn't have a solution? One suggestion has already been given.

Going in another direction, these equations represent lines in the plane. Do these planes intersect, which would give one or possibly an infinite number of solutions, or do they not intersect at all?
 
Math100 said:
k+2=1
1+2k=1
These are the world's easiest equations to deal with. Why are you having trouble?
 
Math100 said:
k+2=1
1+2k=1
The augmented matrix in the OP represents the system of equation
kx+2y=1
x+2ky=1
 
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NFuller said:
The most direct way to declare that a system is inconsistent is by looking for the value of ##k## which causes the determinant of the matrix to be zero. Since this is a fairly simple system however, you could easily pick a value of ##k## which causes one of the rows to be a multiple of the other.
Zero determinant does not necessarily imply that the system is inconsistent.
 
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Got it!
 
  • #10
Math100 said:
Got it!
What did you get for k?
 
  • #11
-1
 
  • #12
How to give award to best answer?
 
  • #13
ehild said:
Zero determinant does not necessarily imply that the system is inconsistent.
Thank you so much!
 
  • #14
You are welcome:oldsmile:
 
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