# What is Gaussian elimination: Definition and 70 Discussions

In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855) although some special cases of the method—albeit presented without proof—were known to Chinese mathematicians as early as circa 179 CE.
To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations:

Swapping two rows,
Multiplying a row by a nonzero number,
Adding a multiple of one row to another row.Using these operations, a matrix can always be transformed into an upper triangular matrix, and in fact one that is in row echelon form. Once all of the leading coefficients (the leftmost nonzero entry in each row) are 1, and every column containing a leading coefficient has zeros elsewhere, the matrix is said to be in reduced row echelon form. This final form is unique; in other words, it is independent of the sequence of row operations used. For example, in the following sequence of row operations (where two elementary operations on different rows are done at the first and third steps), the third and fourth matrices are the ones in row echelon form, and the final matrix is the unique reduced row echelon form.

[

1

3

1

9

1

1

1

1

3

11

5

35

]

[

1

3

1

9

0

2

2

8

0

2

2

8

]

[

1

3

1

9

0

2

2

8

0

0

0

0

]

[

1

0

2

3

0

1

1

4

0

0

0

0

]

{\displaystyle {\begin{bmatrix}1&3&1&9\\1&1&-1&1\\3&11&5&35\end{bmatrix}}\to {\begin{bmatrix}1&3&1&9\\0&-2&-2&-8\\0&2&2&8\end{bmatrix}}\to {\begin{bmatrix}1&3&1&9\\0&-2&-2&-8\\0&0&0&0\end{bmatrix}}\to {\begin{bmatrix}1&0&-2&-3\\0&1&1&4\\0&0&0&0\end{bmatrix}}}
Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss–Jordan elimination. In this case, the term Gaussian elimination refers to the process until it has reached its upper triangular, or (unreduced) row echelon form. For computational reasons, when solving systems of linear equations, it is sometimes preferable to stop row operations before the matrix is completely reduced.

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1. ### Find the solutions of the system, for all λ

Thread moved from technical math section, so there is no homework template. (∀λ∃ℝ) -x + y - z = 1 -2x + 10y + (2λ + 6) = 6 3x + 11y + (λ2+6)z = 5λ - 1 after gaussian elimination I have this: -1 4 -2 | 1 0 1 λ | 2 0 0 λ(λ-1) | 5λ So, for λ=0 ⇒ ∞ solutions, for λ=1...
2. ### B Row reduction, Gaussian Elimination on augmented matrix

Hi! Please, could you help me on how to solve the following matrix ? I need to replace the value 3 on the third line by 0, the first column need to remain zero and 1 for the third column. I'm having a lot of difficulties with this. How would you proceed ? Thank you for your time and help...
3. ### Gaussian Elimination of Singular Matrix with partial pivoting

Part (A): The matrix is a singular matrix because the determinant is 0 with my calculator. Part (B): Once I perform Gauss Elimination with my pivot being 0.6 I arrive at the last row of matrix entries which are just 0's. So would this be why Gauss Elimination for partial pivoting fails for this...
4. ### I Gaussian elimination for homogeneous linear systems

Hi, I ask for a clarification about the following: consider for instance a 10 x 12 homogeneous linear system and perform Gauss elimination for the first 8 unknowns. Suppose you end up with 5 equations in the remaining 12-8 = 4 unknowns (because in the process of the first 8 unknowns elimination...
5. ### I Gaussian elimination for a singular square matrix

Hi, I've the following doubt: consider an homogeneous linear system ##Ax=0## with ##A## a singular square matrix. The resulting matrix attained through Gaussian elimination will be in upper triangular or raw echelon form ? Thanks.
6. M

### Constructing a 3x3 Linear system question

Homework Statement Construct a 3 × 3 example of a linear system that has 9 different coefficients on the left hand side but rows 2 and 3 become zero in elimination. If the right hand sude of your system is <b1,b2,b3> (Imagine this is a column vector) then how many solutions does your system...
7. M

### Reducing a matrix to echelon form

Homework Statement (i) Reduce the system to echelon form C|d (ii) For k = -12, what are the ranks of C and C|d? Find the solution in vector form if the system is consistent. (iii) Repeat part (b) above for k = −18 Homework Equations Gaussian elimination I used here...
8. M

### Finding values to make a linear system consistent

Homework Statement Given the following matrix: I need to determine the conditions for b1, b2, and b3 to make the system consistent. In addition, I need to check if the system is consistent when: a) b1 = 1, b2 = 1, b3 = 3 b) b1 = 1, b2 = 0., b3 = -1 c) b1 = 1, b2 = 2, b3 = 3 Homework...
9. M

### Coefficients that make Gaussian elimination impossible?

Homework Statement Given this matrix: I am asked to find values of the coefficient of the second value of the third row that would make it impossible to proceed and make elimination break down. Homework Equations Gaussian elimination methods I used given here...
10. M

### Why does Gaussian elimination work?

Post moved by moderator from Homework section Hello, I was curious as to why Gaussian elimination works. I know that if we have two ( or more) systems of two (or more) linear equations, we can write then in terms of a matrix. However, what does it mean when I get the identity on the left hand...
11. ### Pivoting (Gaussian elimination)

Homework Statement x1 + 2x2 = 9 , 6x1 + 6x2 -8x3 = 1 , -3x2 +x3 = 0 [/B] Here's a question from my book . I'm asked to find the x1 , x2 and x3 by using the gaussian elimination with pivoting .Homework EquationsThe Attempt at a Solution Pls refer to this thread for the lecturer's method...
12. ### Gaussian elimination (pivoting)

Homework Statement In my book , the author stated that when we do the pivoting , we need to make the element below the leading element = 0 . For example for 3x3 matrix , for the first column , we have to make a11 = max , and make sure that a21 and a31 less than a11 , for the second column , we...
13. ### Solve Gaussian Elimination w/Pivoting: Problem Explained

Homework Statement I am asked to solve this question using pivoting technique ... so , in I , I switched the R1 and R2 so that a11 is max , in III , i switched R9 nad R10 , so that a22 is max . But , in IV , i found that the a31 become -2/9 , which is not 0 anymore , what's wrong with my...
14. ### Gaussian elimination system of Eqs, the case of no solution

Homework Statement system of equations is as follows 4x +2y -2z = 0 2x + y -z= 1 3x +y -2z = 1 Homework EquationsThe Attempt at a Solution [/B] Using gaussian elimination we can multiply mid-eq, by (-2) [[[actually... it is simply a basic equation procedure]]] 2x+y-z=1 |*(-2) = -4x -2y...
15. ### MHB James' question about Gaussian Elimination

We start by writing the system as an augmented matrix \displaystyle \begin{align*} \left[ \begin{matrix} 2 & -2 & 1 & 1 & \phantom{-}4 \\ 4 & -1 & 2 & 3 & -\frac{5}{2} \\ 4 & -4 & 5 & 1 & \phantom{-}8 \\ 0 & \phantom{-}2 & 3 & 4 & -5 \end{matrix} \right] \end{align*} Once we have used...

26. ### How can I improve my GE program using MATLAB?

Homework Statement I am writing a GE program and i think i have that correct. I have a matrix and i am trying to find the correct answer. I am not sure where i am going wrong. Homework Equations The Attempt at a Solution %GE with no pivoting function x=ge_nopivot(A,b)...
27. ### Gaussian elimination: Proof of correctness

I have been looking for a proof of correctness of Gaussian elimination, but alas, without much success. Most online resources explain how to apply the algorithm rather than proving correctness. That said, I have been looking for a proof to the following theorem, which is stated in Friedberg's...
28. ### Modifying a Gaussian Elimination Algorithm to Perform Gauss-Jordan E.

Homework Statement I have an algorithm that implements Gaussian elimination. According to the text, with some modification of the indices and their in the loops, I should be able to have this algorithm perform Gauss-Jordan elimination. I also have to reduce the matrix to reduced row-echelon...
29. ### Solving a System by Gaussian Elimination: Help Needed

Homework Statement for some reason I am having real trouble trying to solve this system. Homework Equations The Attempt at a Solution a=(10,11,4) v1=(2,1,4) v2=(-1,-2,1) v3=(3,3,-1) the qstn does a belong to the span (v1,v2,v3)? so I do gauss elim but I keep running...
30. ### Real Life Applications of Infinite Solutions with Gaussian Elimination

I would normally use Gaussian ELimination to solve a linear system. If we have more unknowns than equations we end up with an infinite number of solutions. Are there any real life applications of these infinite solutions? I can think of solving puzzles like Sudoku but are there others?
31. ### Finding a basis in ImT using Gaussian Elimination

Homework Statement $$\begin{pmatrix} -1&3&0\\ 2&0&-1\\ 0&-6&1 \end{pmatrix}$$ Finding the ImT basis of this The Attempt at a Solution I got it down to $$\begin{pmatrix} 1&0&-1/2\\ 0&1&1/6\\ 0&0&1 \end{pmatrix}$$ I know that by the principle of having pivots as the only non-zero...
32. ### Applying Gaussian Elimination to a Matrix - Understanding the Correct Method

Homework Statement Apply Gaussian elimination to the following matrix 2 -3 0 3 4 -5 1 7 2 -1 -3 5 I understand how to get the answer. The Attempt at a Solution My question is why doesn't the following method work: Switch Z over into the first column, move row 1 to row 3, then exchange rows...
33. ### Gaussian Elimination rules - apply to rows only?

Homework Statement In this video; the uploader states at 0:47 that there are three rules. 1. Can swap rows 2. Can multiply a row by a constant 3. Can add a multiple of one row to another Do these rules apply only to rows? Can they be applied to columns? The uploader has not stated...
34. ### Understanding Gaussian Elimination Matrices for Solving Linear Equations

Homework Statement What is the Gaussian elimination matrix for the vector for the vector v=(2,-10,16,2)? The Attempt at a Solution I don't understand the question. What is a "Gaussian elimination matrix for a vector?" I know that Gaussian elimination is a method for solving a system of...
35. ### Solving via Gaussian Elimination

Homework Statement Solve the following system of equations using Gaussian Elimination: -2a+5c=1 a+2b-c=2 3a-2b=3 Homework Equations N/A The Attempt at a Solution Augmented matrix (-2 0 5 | 1) R2 → R2 +R1/2 (1 2 -1 | 2) R3 → R3 - 3R2 (3 -2 0 | 3) (-2 0 5 | 1) Swap R2 and...
36. ### Gaussian Elimination, is this method OK?

Homework Statement A= <1, -3, -1, 1> <2, -5, 0, 1> <-3, 5, -6, 3> What I did was Row2 = 2*Row1 - Row2 which renders Row2 as: <0, -1, -2, 1> However in the solutions, Row2 was given as: <0, 1, 2, -1>, which appears to be R2 = -2R1 + R2 I'm guessing it makes no real difference...
37. ### Online tools for Gaussian elimination / LU decomposition ?

I can not find any useful online tool that solves you a system of equations using Gaussian elimination and LU decomposition. So just a system like: -3X+4X+9X+4x=-2 9X+2X+1X-5X=2 etc. Just about 4 lines. several lines. So basically an online tool where you can just plug in the...
38. ### Gaussian Elimination Solve for variable

Homework Statement Determine the value of k for which the system x + y + 2z = 3 x + 2y -4z = -3 2 + 8 + kz = -29 has no solutions. Homework Equations None The Attempt at a Solution Converting to a matrix 1 1 2 3 1 2 -4 -3 2 8 k -29 r2=r2-r1 1 1 2 3 0 1 -6 0 2 8 k -29...
39. ### What is exactly pivoting ( in Gaussian elimination)?

In terms of programming. Like, let's say you have F1+F2+F3+15=0 and you move the 15 to the other side, just as you're supposed to, which would be F1+F2+F3= -15. Is that what they mean by pivoting? The code moves the 15? thanks
40. ### How Does Gaussian Elimination Help Solve Systems of Equations?

Homework Statement Using Gaussian Elimination: 3x - 2 y = 5 6x-4y =7The Attempt at a Solution [ 3 -2 5] [6 -4 7] * top row by 1/3 [1 -2/3 5/3] * Top row by -6, add to bottom row: [0 0 -3] So I get this: [1 -2/3 5/3] [0 0 -3] How can 0y equal -3?
41. ### Code for Gaussian Elimination

I have this code for a Gaussian Elimination: As you can see, the input is A and b. I'm trying to figure out how I can make this code work for this Gaussian Elimination: http://img189.imageshack.us/f/50468910.png/ It's 17 rows. The code can do 17 rows. But some of the equations aren't in the...
42. ### Solving Gaussian Elimination Homework

Homework Statement http://img189.imageshack.us/i/50468910.png/ Homework Equations The Attempt at a Solution I'm wondering how I do this? I don't really know. Supposed to be 17 rows. How many columns and how many each row? No clue! And how do I get the numbers for the matrix here?
43. ### Gaussian Elimination Solution to the 2D Poisson Equation

I am trying to use Gaussian elimination to solve the 2D poisson equation. I've done this for the 1D problem without problems, but for some reason my solution for the 2D problem is incorrect; it looks something like the correct solution but it's as if the resulting field were cut in half, so...
44. ### Finding Perpendicular Vectors in R^4 Using Gaussian Elimination

Find all vectors in R^4 that are perpendicular to the three coloumn vectors [1 1 1 1], [1 2 3 4], [1 9 9 7] sorry i don't know how to make them verticle guassian elimination or Ax=b I know that the dot product must equal 0 but I am not sure how to solve it

Got it
46. ### Gaussian Elimination: Solving Systems of Linear Equations

Is it correct that the Gaussian elimination procedure is used in computer software to solve systems of linear equations?
47. ### Matlab Gaussian elimination with partial pivoting

Homework Statement Hi all, I'm writing a program to solve a system of linear algebraic equations using the method of Gaussian elimination. The only thing I can't figure out is how to perform the actual pivot. In the %Forward elimination nest, I can't figure out how I am supposed to find the...
48. ### Gaussian Elimination: Translating text into equations

Hello :smile: I have a homework question regarding Gaussian Elimination, where I am supposed to use information in a block of text to get equations, and form an augmented matrix. There are 4 unknowns, but I can only seem to get 3 equations and 3 "checks", to see if the values I get are...
49. ### Exploring the Impact of Gaussian Elimination: A Comprehensive Analysis

My linear algebra teacher is strange. This is proof, he wants us to write a 4 page essay about the strengths and weaknesses of the Gaussian Elimination method to solve systems of equations covering these four areas: accuracy, student understanding, preparation for higher-level mathematics...
50. ### Solving linear systems + gaussian elimination

Homework Statement http://img.rflz.net/problems.jpg Homework Equations The Attempt at a Solution Question 1 x4 = 1 + 4x5 but from there I am not sure where to go or if there is even a solution? Question 2 I eventually got the matrix down to this: 1 2 -3 1 4 1 0 1 1 0...