Augmented Matrix in Echelon Form - One Unique Solution?

  • Thread starter L²Cc
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In summary, the conversation is about a question regarding a 3 by 3 matrix in echelon form. The question is clarified to state that the last row has two zeros on the far right and two real numbers on the left. It is confirmed that this indicates there is one unique solution, z=3. The issue with the template not appearing in the writing box is also addressed.
  • #1
L²Cc
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Before anyone comments, the template which you provide does not appear in the writing box for some reason...thus, I'll have to write out my question differently!

When a 3 by 3 matrix is in echelon form, what does it mean when the last row has two zeros on the far right side and two real numbers on the other side so that it looks like this: 0 0 1|3
I know when the entire row consists of zeros, the unknowns have no solutions, when there is one really number and the answer is zero the matrix has infinitely many solutions...I'm guessing the above implies that the matrix has one unique solution? right?
 
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  • #2
L²Cc said:
Before anyone comments, the template which you provide does not appear in the writing box for some reason...thus, I'll have to write out my question differently!

When a 3 by 3 matrix is in echelon form, what does it mean when the last row has two zeros on the far right side and two real numbers on the other side so that it looks like this: 0 0 1|3

Note that the zeros are on the *left* hand side of the row!

Anyway, yes there is one solution, namely z=3 (supposing you are using coordinates x,y,z, that is!)
 
  • #3
oops, human error! Thank you Cristo. Had to clarify...
 
  • #4
L²Cc said:
Before anyone comments, the template which you provide does not appear in the writing box for some reason...thus, I'll have to write out my question differently!
Are you using the Nexus skin? If this is the case, the template won't show up because it hasn't been implemented in that skin yet.
 

1. What is an augmented matrix in echelon form?

An augmented matrix in echelon form is a type of matrix used in linear algebra to represent a system of linear equations. It is arranged in a triangular shape with zeros below the main diagonal, and the rightmost column contains the constants from the equations. This form makes it easier to solve the system of equations and find the unique solution.

2. What does it mean for an augmented matrix to have one unique solution?

A system of linear equations represented by an augmented matrix can have either no solution, infinite solutions, or one unique solution. When an augmented matrix is in echelon form and has one unique solution, it means that the system of equations has a single solution that satisfies all the equations in the system. In other words, there is only one set of values for the variables that can make all the equations true.

3. How do you know if an augmented matrix has one unique solution?

An augmented matrix has one unique solution if it is in echelon form and has no zero rows. This means that all the variables in the system of equations can be solved for, and there are no contradictions or redundancies in the equations. Additionally, in an augmented matrix with one unique solution, the number of variables must be equal to the number of non-zero rows in the matrix.

4. Can an augmented matrix have one unique solution if it is not in echelon form?

No, an augmented matrix must be in echelon form to have one unique solution. If the matrix is not in echelon form, it may have either no solution or infinite solutions. This is because the triangular structure of an echelon form matrix allows for a clear and systematic way of solving the equations, whereas a non-echelon form matrix may have redundancies or contradictions that make it impossible to find a unique solution.

5. What is the importance of finding the augmented matrix in echelon form with one unique solution?

Finding the augmented matrix in echelon form with one unique solution is important because it allows us to efficiently solve systems of linear equations without having to manually solve each equation. This form also helps us identify if a system has no solution or infinite solutions, and it provides a systematic way of finding the unique solution. Additionally, the echelon form allows us to easily perform operations like row reduction to further simplify the matrix and solve the system of equations.

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