Consistency of A System of Linear Equations

In summary, the conversation discussed the concept of consistency in a system of linear equations. It was mentioned that a system is consistent if it can be put into triangular form without any contradictions. The question was raised as to why being in triangular form implies consistency. It was suggested to think about what consistency means in relation to a triangular matrix and whether all triangular matrices can be diagonalized.
  • #1
Bashyboy
1,421
5
Hello everyone,

I was just solving a problem in which I had to determine the system of linear equations were consistent. Evidently, if a system of linear equations is capable of being put into triangular form, with no contradictions present, then it must consistent. My question is, why is that so, why does being in triangular form imply consistency?
 
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  • #2
You should be able to work it out - what does it mean for the system of equations to be "consistent"?

If the system is represented by a triangular matrix, what is the form of the corresponding equations?
Would these be consistent?

Can you see that a diagonal matrix means consistency?
Are there any triangular matrixes that cannot be diagonalized?
 

Related to Consistency of A System of Linear Equations

1. What is the definition of consistency of a system of linear equations?

The consistency of a system of linear equations refers to whether or not there exists at least one solution that satisfies all of the equations in the system.

2. How can I determine if a system of linear equations is consistent?

A system of linear equations is consistent if and only if the number of equations is equal to the number of unknown variables, and the system is solvable. This can be determined by using Gaussian elimination or other methods of solving systems of equations.

3. What does it mean if a system of linear equations is inconsistent?

If a system of linear equations is inconsistent, it means that there is no solution that satisfies all of the equations in the system. This can occur when the equations are contradictory or when there are more equations than unknown variables.

4. Can a system of linear equations be both consistent and inconsistent?

No, a system of linear equations cannot be both consistent and inconsistent. It can only be one or the other.

5. How does the consistency of a system of linear equations affect its solutions?

If a system of linear equations is consistent, it means that there exists at least one solution. However, if a system is inconsistent, there will be no solution. Inconsistent systems may also have an infinite number of solutions, depending on the specific equations.

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