For questions on or related to the technique of Gaussian elimination, used in solving systems of linear equations.

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Why use Gauss Jordan Elimination instead of Gaussian Elimination, Differences

Why use Gaussian Elimination instead of Gauss Jordan Elimination and vice versa for solving systems of linear equations? What are the differences, benefits of each, etc.? I've just been solving ...
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Gaussian Elimination and Matrix [on hold]

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. \begin{cases} 4x - y + 3z = 12 \\ x + 4y + 6z = -32 \\ 5x + 3y + 9z = 20 ...
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1answer
28 views

Linear Transforms & Matrices

$T:R^4 -> R^3$ Linear Transform This matrix is $[T]_{B2}^{B1}$ = A =\begin{pmatrix}1&2&3&4\\1&4&0&2\\2&2&9&10\end{pmatrix} After elimination we get: ...
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28 views

Solving Linear equation Gaussian method of elimination [closed]

Solving linear equation Gaussian method of elimination $$\left\{ \eqalign{ -x-y & =7 \\ 4x-y & =-3 } ...
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22 views

Linear Algebra - elimination and linear systems

By given this matrix: \begin{pmatrix}1&1&1&0\\2&3&k&1\\3&k&5&1\end{pmatrix} I need to find, what are the values of k the system has infinity/single/no solution. So ...
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23 views

Gaussian Elimination diagonal elements

For a matrix A, assume that B is the upper triangular matrix after applying Gaussian elimination on A. I want to calculate only the diagonal elements of the output matrix B in terms of $A_{ij}$ (the ...
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22 views

Gaussian elimination vs. Jacobi iteration

How can I determine which of the matrix solver is faster for a given set of equations: Gaussian elimination or Jacobi iteration? In case, I have a banded matrix, is it advisable to use LU ...
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4answers
97 views

Can you use row and column operations interchangeably?

Is it possible to use row and column operations "at the same time" on a matrix $A$? So, for example, first subtracting $row_1$ from $row_2$, and then choosing to multiply $column_3$ by a constant $c$? ...
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1answer
29 views

Gaussian elimination easily?

Let $$\left( {\matrix{ 1 & 0 & { - 3} \cr 0 & 2 & {\lambda + 3} \cr 0 & 0 & {5 - {\lambda ^2}/2 - 3\lambda /2} \cr } \left| {\matrix{ { - 3} \cr 2 ...
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1answer
28 views

Gauss-Jordan Method

I keep getting the wrong set of solutions can someone help me. I know that when using the Gauss-Jordan method, the rules that I must follow can be applied in a variety of different procedures then why ...
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1answer
25 views

Calculating the rank of two given matrices

I read somewhere that the rank of a matrix is the number of its nonzero rows or columns after Gaussian elimination. In the following matrices, how should I know Gaussian elimination is done? They are ...
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1answer
28 views

Specific system of equations with multiplications

I'm facing a math problem that I thought easy, but I'm stuck with a solution that doesn't seem optimal. The problem is the following : I have "registers" which are the expanded representation of ...
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2answers
65 views

how to solve a 4x5 matrix with guassian elimination or gaussian-jordan

$$ \left[ \begin{array}{ccccc} 2 & -1 & 3 & 4 & 9 \\ 1 & 0 & -2 & 7 & 11 \\ 3 &-3 & 1 & 5 & 8 \\ 2 & 1 & 4 & 4 & 10 \\ ...
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2answers
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Get variables with Matrix

I try to get the variables for this equation: $$\begin{cases} 6x_1 + 4x_2 + 8x_3 + 17x_4 &= -20\\ 3x_1 + 2x_2 + 5x_3 + 8x_4 &= -8\\ 3x_1 + 2x_2 + 7x_3 + 7x_4 &= -4\\ 0x_1 + 0x_2 + 2x_3 ...
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1answer
21 views

Get variables with Gauß

I try to find the solutions for the following expression with the Gauß Formula: $$1x_1-1x_2+3x_3=0$$ $$2x_1 + 3x_2 - 1x_3 = 0$$ $$3x_1+7x_2-5x_3 =0$$ So i started: $ \begin{pmatrix} ...
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2answers
32 views

Method for Finding Matrix-Inverse Through Gauss-Jordan?

When trying to find the inverse of the n$\times$n matrix $A$, one way of going about it is by solving $AX=I$, wherein $I$ is the n$\times$n identity matrix, and $X$ is some n$\times$n matrix which is ...
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4answers
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Why do the addition of linear equations all pass through the same point

So I'm doing linear algebra right now and I have a question regarding addition of equations as part of Gauss' elimination algorithm. I understand why it's possible, as the LHS of one equation can be ...
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1answer
54 views

Is it possible for a system of equations to have a non-zero determinant and no solution at the same time?

I am quite confused by the solution I was given for the following problems: a) Solve the following system of equations using Gauss elimination only: $2x - y = 5$ $-x + 2y = -4$ $3x - y = -1$ b) ...
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1answer
27 views

Row reducing a matrix with determined pivots through Gauss-Jordan

In Algebra we've been given this matrix: $P=\begin{pmatrix} 3 & 2 & 6 & 10 \\ 8 & 4 & 9 & 5 \\ 7 & 3 & 12 & 4 \end{pmatrix}$ I'm asked to row-reduce it with ...
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1answer
39 views

solve linear system using gaussian elimination

I want to solve a linear system of the form Ax=b. First of all I create the augmented matrix (A|b). I apply some elementary row operations and i obtain the REF form of A. After than, I do not know ...
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2answers
29 views

finding determinant for matrix using upper triangle method!

so Here an example for matrix that I'm trying to evaluate its determinant! | 1 3 2 1| | 0 1 4 -4| | 2 5 -2 9| | 3 7 0 1| when applying first row operation i get | 1 3 2 1| | 0 1 ...
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0answers
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Growth factor for Gaussian Elimination WITHOUT partial pivoting

A = $\begin{bmatrix} 1 & 7 & -11 \\ 4 & 29 & -50 \\ 6 & 49 & -107 \end{bmatrix}$ , after computing the LU factorization without partial pivoting I have L = ...
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2answers
36 views

Finding the inverse of a matrix by Gaussian elimination

I spent last hours trying to figure out how to solve the inverse matrix to this matrix: $$\begin{pmatrix} 2 &-3 & 1 \\ 1 & 2 &-1 \\ 2 & 1 & 1 \end{pmatrix}$$ The correct ...
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0answers
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What changes where made on this Gaussian-Elimination?

in the Internet I have found the following use of the Gaussian Elimination method: $z \in \mathbb{R}, \ n\in\mathbb{N}, n \ge 2$ and $\begin{pmatrix} z & 1 & \dots & 1 & 1 \\ 1 & ...
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1answer
35 views

What is the canonical basis of a dualspace in $\mathbb{R}^3$?

I have the following: Consider the basis $$B := \{\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \}$$ of the ...
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2answers
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$3$ lines $4$ variables linear equation gaussian

So I'm currently taking a Linear Algebra class and am stuck on a problem. I have the equations: $$\begin{cases}\begin{align}&x + 2y - z + 3t = 3\\ &2x + 4y + 4z + 3t = 9\\ &3x + 6y - z ...
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1answer
78 views

Gauss-Jordan: Effect of column pivoting on result matrix

When I implement a Gauss-Jordan algorithm I can either have a 1 column result matrix or a multi-column result matrix (I mean the right hand side of the augmented matrix). The first case would be the ...
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1answer
57 views

Linear system of equations and multiple linear regression: Numerical solving

I am currently implementing a test procedure for data, namely a linear form of the Kramers-Kronig relations (paper here: http://jes.ecsdl.org/content/142/6/1885.abstract). This includes solving a ...
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1answer
96 views

Find the values of $k$ that make this system inconsistent, with unique solution, and with infinite solutions.

I've learned to find the solutions to linear systems using Gaussian Elimination. Moving on, I've found a new kind of exercise I hadn't done before: Find the values for $k$ that make this system: ...
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42 views

What did I do wrong with Gaussan Elimination for $\begin {cases} x + 5y + 11z = -5\\ 2x + 3y + 8z = 4\\ -x + 2y + 3z = -9 \end {cases}$?

Having problems with this one using Gaussian Elimination. Find the solutions for the linear equation system: $$\begin {cases} x + 5y + 11z = -5\\ 2x + 3y + 8z = 4\\ -x + 2y + 3z = -9 \end ...
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Row operations that change similarity class

Let $\mathbb{K}$ be a field and $A\in \mathcal{M}_{n\times n}(\mathbb{K})$ be a matrix. Which row operations on $A$ do not change its similarity class?
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Does non linear transform of Gaussain Random Variables results in Gaussian?

I have a question, if we do a non linear transformation on Gaussian random vector, will it give us Gaussian as a result? If No which techniques can we use to make sure the result is finally gaussian. ...
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5answers
45 views

How to solve this homogeneous system, with a missing column?

Find the solution set of triplets $(x,y,z)$ that fulfil this system using Gauss-Jordan: $$\begin {cases} -x + 2z = 0\\ 3x - 6z = 0\\2x - 4z = 0\end {cases}$$ First of all, I don't see any ...
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Using Gauss-Jordan for an infinite-solutions system

I'm starting to get the hang of this Gauss-Jordan stuff - well, I have never done a system with infinite solutions, so I decided to try this one. You can scroll to the bottom instead to see my doubts ...
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4answers
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Are there no solutions for $\begin {cases} 2x+4y = 6\\ 3x+6y = 5\end {cases}$?

I'm trying to solve an equation system using Gauss-Jordan. $$\begin {cases} 2x+4y = 6\\ 3x+6y = 5\end {cases}$$ So, first, the augmented matrix: \begin{bmatrix} 2&4&5\\ 3&6&6\\ ...
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1answer
71 views

Can all equation systems be reduced to the identity matrix?

I'm trying to learn about solving equation systems using the Gauss-Jordan method. So, you have to convert the equation system to a matrix, and then reduce it to the identity. When you transform it to ...
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2answers
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Row reduced matrix $\Leftrightarrow$ vectors (rows) are linearly independent.

Let $A$, a row-reduced matrix (after applying Gaussian elimination). Show that all rows which are different from $V_0$ (zero vector), are linearly independent. We learned this as sort of an ...
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1answer
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Finding the number of solutions to two equations

I have a question: For the following system of linear equations, using Gaussian elimination, decide whether it has at least one solution. If it does, represent the general solution as an affine map ...
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50 views

Gauss Jordan Elimination [closed]

Apply Gauss-Jordan elimination to the following system and determine the general solution if it exists. $x+y+2z+3u+4v=0\\2x+2y+7z+11u+14v=0\\3x+3y+6z+10u+15v=0.$
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Gauss Elimination - Diagonal dominant matrices don't need row changes

I was asked to prove the following statement: let $A$ be an $n$ by $n$ matrix with real entries such that $\forall k \in \mathbb N, k\leq n$: $$\sum_{i \neq k} |A_{i,k}| < |A_{kk}|$$ Show that if ...
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1answer
62 views

LU factorization problem - Writing a code, don't understand partial pivoting

I'm trying to write a matlab code for the following question: The program gets a matrix $A$ (lets say square matrix) and it returns $P,L,U$ such that $PA=PLU$ and $P$ is the permutation matrix, the ...
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1answer
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Not getting the right answer for a matrix in reduced column echelon form.

This question is NOT homework (I'm not reading the referenced textbook as part of a class). Introduction to Linear Algebra by Donald J. Wright has this question in section 1.7: Find a matrix in ...
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3answers
124 views

Find a second degree polynomial that goes through 3 points

I am having trouble calculating the quadratic curve $f(x)$ that goes through 3 points; for example a curve that goes through $A(1,3), B(-1,-5), and C(-2,12)$. I can only guess that the curve is ...
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1answer
267 views

How to solve a 3x4 matrix has no solution, a unique solution, and infinite solutions??

The system is : $$ \begin{matrix} 1 & -4 & 6 & a & | & 0 \\ -2 & 5 & -4 & -1 & | & b \\ 1 & -10 & 22 & 8 & | & c \end{matrix} $$ After ...
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1answer
88 views

Solve $A\cdot x = b$ by Naïve Gaussian elimination

The following is a homework problem: Let: $$A = \left[\matrix{4& 2& -5 &1\\ -8& 0& 9& 7\\ -32& -4& 43& 18\\ 24 &4 &-22 &-8\\} \right]$$ $$b = ...
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0answers
54 views

Measuring Reasonableness

Measuring Reasonableness I'm working on a problem I refer to as "measuring reasonableness". My asks are along two lines: on one hand, I'm struggling with a specific problem on my current approach and ...
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1answer
58 views

Prove Gaussian Elimination Preserves this Matrix Property

Prove or disprove that if a matrix has the property $0 \neq |a_{ii}| \leq \sum_{\substack{j=1 \\j \neq i }}|a_{ij}|$ Then Gaussian elimination without pivoting will preserve this property I have ...
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1answer
199 views

Gauss-Jordan Elimination to solve for variables

I have the following linear system: $$x + 2y - 3z = 4$$ $$3x - y + 5z = 2$$ $$4x + y + (s^2 - 14)z = s+2$$ Im trying to solve for $s$ to figure out how many solutions it has (if any). I know how to ...
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2answers
55 views

Canonical form for $x^2+y^2+z^2+xy+xz+yz$, using Gauss

I need to find a canonical form for the following equation, throughout the Gauss method. $$x^2+y^2+z^2+xy+xz+yz$$ And I'm stuck at this point, because even if I continue to creeate $(a+b+c)^2$ it ...
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4answers
38 views

On which terms $rank(A)=3$?

$$\left( {\begin{array}{*{20}{c}} a & b & {a + b} \\ {2a} & {a + b} & {a - b} \\ a & a & {2a - b} \\ \end{array}} \right) \simeq \left( {\begin{array}{*{20}{c}} ...