Chapters

Row reduction or Gauss method is a linear algebra technique used to solve systems of linear equations and find the row echelon or reduced row echelon form of a matrix, simplifying calculations.

In this series of exercises, we will explore various problems involving Gaussian reduction, giving you the opportunity to develop your skills in this important mathematical concept. Let's start practicing!

The best Mathematics tutors available
Reza
5
5 (50 reviews)
Reza
$40
/h
Gift icon
1st lesson free!
Jose
5
5 (35 reviews)
Jose
$35
/h
Gift icon
1st lesson free!
Josiah
5
5 (108 reviews)
Josiah
$30
/h
Gift icon
1st lesson free!
Lyle
5
5 (40 reviews)
Lyle
$35
/h
Gift icon
1st lesson free!
Davayne
5
5 (120 reviews)
Davayne
$30
/h
Gift icon
1st lesson free!
Sofia
5
5 (64 reviews)
Sofia
$65
/h
Gift icon
1st lesson free!
Joe
4.9
4.9 (35 reviews)
Joe
$25
/h
Gift icon
1st lesson free!
Fadil
5
5 (43 reviews)
Fadil
$35
/h
Gift icon
1st lesson free!
Reza
5
5 (50 reviews)
Reza
$40
/h
Gift icon
1st lesson free!
Jose
5
5 (35 reviews)
Jose
$35
/h
Gift icon
1st lesson free!
Josiah
5
5 (108 reviews)
Josiah
$30
/h
Gift icon
1st lesson free!
Lyle
5
5 (40 reviews)
Lyle
$35
/h
Gift icon
1st lesson free!
Davayne
5
5 (120 reviews)
Davayne
$30
/h
Gift icon
1st lesson free!
Sofia
5
5 (64 reviews)
Sofia
$65
/h
Gift icon
1st lesson free!
Joe
4.9
4.9 (35 reviews)
Joe
$25
/h
Gift icon
1st lesson free!
Fadil
5
5 (43 reviews)
Fadil
$35
/h
Gift icon
1st lesson free!
Let's go

System of 3 Equations with 2 Variables

1

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

2

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

3

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

4

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

5

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

System of 2 Equations with 3 Variables

1

Solution

We write in matrix form

We apply the Gauss method

The system is consistent and indeterminate

We perform a parameterization of the solution using . In this way, the second equation becomes:

That is, .

 

On the other hand, the first equation becomes , which when solving for gives us:

That is, .

2

Solution

We write in matrix form

 

 

We apply the Gauss method

 

The system is consistent and indeterminate

We perform a parameterization of the solution using . In this way, the second equation becomes:

 

 

That is, .

 

On the other hand, the first equation becomes , which when solving for  gives us:

 

 

That is, .

3

Solution

We write in matrix form

 

 

We apply the Gauss method

 

The system is consistent and indeterminate

We perform a parameterization of the solution using . In this way, the second equation becomes:

 

 

That is, .

 

On the other hand, the first equation becomes , which when solving for gives us:

 

4

Solution

We write in matrix form

 

 

We apply the Gauss method

 

The system is consistent and indeterminate

We perform a parameterization of the solution using . In this way, the second equation becomes:

 

 

That is, .

 

On the other hand, the first equation becomes , which when solving for gives us:

 

5

Solution

We write in matrix form

 

 

We apply the Gauss method

 

The system is consistent and indeterminate

We perform a parameterization of the solution using . In this way, the second equation becomes:

 

 

That is, .

 

On the other hand, the first equation becomes , which when solving for gives us:

 

System of 3 Equations with 3 Variables with Similar Coefficients

1

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

2

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

3

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

4

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

5

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

System of 3 Equations with 3 Variables

1

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

The system is consistent and determined

2

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

The system is consistent and determined

3

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

4

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

5

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

Verify if the Following System Is Determined or Indeterminate

1

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and indeterminate

2

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

The system is consistent and determined

3

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

4

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and determined

5

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

The system is consistent and indeterminate

System of 4 Equations with 4 Variables

1

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

The system is consistent and indeterminate

We have that the system is underdetermined since the last row was canceled. We will parameterize the solution using . The second equation becomes:

From here we proceed to express in terms of using the third equation, which becomes:

Finally, we use the first equation to express in terms of :

That is,

2

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

The system is consistent and determined

3

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

The system is consistent and determined

4

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

The system is consistent and determined

5

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

The system is consistent and indeterminate

 
We have that the system is underdetermined since the last row was canceled. We will parameterize the solution using . The third equation becomes:

From here we proceed to express in terms of using the second equation, which becomes: Finally, we use the first equation to express in terms of :

That is,

Verify the Indetermination of the System of 4 Equations

1

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

The system is consistent and indeterminate

We have that the system is underdetermined since the last row was canceled. We will parameterize the solution using . The second equation becomes:

From here we proceed to express in terms of using the third equation, which becomes:

Finally, we use the first equation to express in terms of :

Thus

2

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

The system is consistent and indeterminate

We have that the system is underdetermined since the third row was canceled. We will parameterize the solution using . The second equation becomes:

From here we proceed to express in terms of using the third equation, which becomes:

Finally, we use the first equation to express in terms of :

Thus

3

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

The system is inconsistent

4

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

The system is consistent and indeterminate

We have that the system is underdetermined since the fourth row was canceled. We will parameterize the solution using . The third equation becomes:

From here we proceed to express in terms of using the second equation, which becomes:

Finally, we use the first equation to express in terms of :

5

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

The system is consistent and determined

 

Solve the System of 3 Equations and 5 Variables

1

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

The system is consistent and determined

 
We will parameterize the solution using . The third equation becomes:

From here we proceed to express in terms of using the second equation, which becomes:

Finally, we use the first equation to express in terms of :

Thus

2

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

The system is consistent and indeterminate

We will parameterize the solution using . The third equation becomes:

From here we proceed to express in terms of using the second equation, which becomes:

Finally, we use the first equation to express in terms of :

Thus

3

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

The system is consistent and indeterminate

We will parameterize the solution using . From the third equation we obtain:

From here we proceed to express in terms of using the second equation, which becomes:

Finally, we use the first equation to express in terms of :

4

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

The system is consistent and indeterminate

We will parameterize the solution using . From the third equation we obtain:

From here we proceed to express in terms of using the second equation, which becomes:

Finally, we use the first equation to express in terms of :

5

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

The system is consistent and indeterminate

We will parameterize the solution using . From the third equation we obtain:

From here we proceed to express in terms of using the second equation, which becomes:

Finally, we use the first equation to express in terms of :

Solve the System of 4 Equations with 3 Variables

1

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

 

The system is inconsistent

2

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

 

The system is consistent and determined

 

3

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

 

The system is inconsistent

4

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

 

The system is inconsistent

5

Solution

We write in matrix form

 

 

We apply the Gauss method

 

 

 

 

 

The system is inconsistent

Did you like this article? Rate it!

5.00 (2 Note(n))
Loading...

Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.