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.
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...
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...
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...
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
I'm not sure where you're getting the idea that z = 2, as this is not correct.
I'm assuming this is to be done without pivoting...
Set up your augmented matrix:
$\displaystyle \begin{align*} \left[ \begin{matrix} 2 & \phantom{-}1 & -3 & -5 \\ 1 & -1 & \phantom{-}2 & 12 \\ 7 & -2 & \phantom{-}...
Homework Statement
I started with the following equations :
3w+2x+3y+z=76
w+2x+2y+z=59
w/2+x+y+z/4=21
Homework Equations
Gaussian elimination
The Attempt at a Solution
So I got the following after applying Gauss' method :
3w+2x+3y+z=76
-4x-3y-2z=-101
3/2z=51
which gives z=34
Then I get...
I am attempting to find the determinant using gaussian elimination for the following matrix [4 3 2; 1 7 8; 3 9 3].
I have begun by attempting to form zeros below the diagonal. The answer should be -165, however i keep getting values up to -665. I have tried a variety of times and keep getting...
I attempting to find the determinant using gaussian elimination for the following matrix [1 2 3; 3 2 2; 0 9 8].
I have begun by attempting to form zeros below the diagonal. My first row operation was to make row 2 equal to 3(row 1) - row 2. This gives me [1 2 3; 0 4 7; 0 9 8] . I think i am...
If I switch 2 rows, do I have to multiple by -1 each time?
For example, I have
If I switch row 2 and 3, will it become this:
Or this?
Each time I make a switch, do I have to also put a negative sign?
Edit: Not really related to Gaussian elimination, but this is from a Gaussian...
Use Gaussian elimination with back substitution to solve the following system:
x1+x2+x3=1,
x1+2x2+2x3=1,
x1+2x2+3x3=1.
The answer is (1, 0, 0) and I know how to solve the problem but I just don't know if I should use bracket or the big parentheses for this type of problem when I solve...
Homework Statement
When solving 2 equations using gaussian elimination, can we divide one equation by the other?
Can you help me find where I went wrong?
Homework Equations
4x + 2y = 14
2x-y=1
The Attempt at a Solution
I would like to check if what I have done is correct. Please, any input is appreciated.
**Problem statement:** Consider a non-singular matrix $A_{nxn}$. Construct an algorithm using Gaussian elimination to find $A^{-1}$. Provide the operation counts for this algorithm.
**My attempted algorithm...
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)...
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...
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...
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...
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?
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...
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...
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...
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...
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...
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...
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...
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
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?
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...
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?
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...
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
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...
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...
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...
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...